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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Practical sequences Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. 78. Population growth When a biologist begins a study, a colony of prairie dogs has a population of 250. Regular measurements reveal that each month the prairie dog population increases by 3%. Let pn be the population (rounded to whole numbers) at the end of the nth month, where the initial population is p0 = 250.Practical sequences Consider the following situations that generate a sequence.. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. 79. Radioactive decay A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mn be the mass of the radioactive material at the end of the nth decade, where the initial mass of the material is M0 = 20 g.Consumer Price Index The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year 1984. Assume the CPI has increased by an average of 3% per year since 1984. Let cn be the CPI n years after 1984, where c0 = 100.Drug elimination Jack took a 200-mg dose of a painkiller at midnight. Every hour, 5% of the drug is washed out of his bloodstream. Let dn be the amount of drug in Jacks blood n hours after the drug was taken, where d0 = 200 mg.A square root finder A well-known method for approximating c for a positive real number c consists of the following recurrence relation (based on Newtons method; see Section 4.8). Let a0 = c and an+1=12(an+can),forn=0,1,2,3,. a. Use this recurrence relation to approximate 10. How many terms of the sequence are needed to approximate 10 with an error less than 0.01? How many terms of the sequence are needed to approximate 10 with an error less than 0.0001? (To compute the error, assume a calculator gives the exact value.) b. Use this recurrence relation to approximate c, for c = 2, 3,, 10. Make a table showing the number of terms of the sequence needed to approximate c with an error less than 0.01.Fixed-point iteration A method for estimating a solution to the equation x = f(x), known as fixed-point iteration, is based on the following recurrence relation. Let x0 = c and xn+1 = f(xn), for n = 1, 2, 3, ... and a real number c. If the sequence {xn}n=0 converges to L, then L is a solution to the equation x = f(x) and L is called a fixed point of f. To estimate L with p digits of accuracy to the right of the decimal point, we can compute the terms of the sequence {xn}n=0 until two successive values agree to p digits of accuracy. Use fixed-point iteration to find a solution to the following equations with p = 3 digits of accuracy using the given value of x0. 79. x = cos x; x0 = 0.8Fixed-point iteration A method for estimating a solution to the equation x = f(x), known as fixed-point iteration, is based on the following recurrence relation. Let x0 = c and xn+1 = f(xn), for n = 1, 2, 3, ... and a real number c. If the sequence {xn}n=0 converges to L, then L is a solution to the equation x = f(x) and L is called a fixed point of f. To estimate L with p digits of accuracy to the right of the decimal point, we can compute the terms of the sequence {xn}n=0 until two successive values agree to p digits of accuracy. Use fixed-point iteration to find a solution to the following equations with p = 3 digits of accuracy using the given value of x0. 80. x=x3+120;x0=5Classify the following sequences as bounded, monotonic, or neither. a. {12,34,78,1516,...} b. {1,12,14,18,116,...} c. {1, 2, 3, 4, 5, ...} d. {1, 1, 1, 1, ...}Describe the behavior of {rn} in the cases r = 1 and r = 1.3QCWhich sequence grows faster: {ln n} or {n1.1}? What is limnn1,000,000en?Give an example of a nonincreasing sequence with a limit.Give an example of a nondecreasing sequence without a limit.Give an example of a bounded sequence that has a limit.Give an example of a bounded sequence without a limit.For what values of r does the sequence {rn} converge? Diverge?Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 44. {0.2n}Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 49. {1.00001n}Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 51. {(2.5)n}Geometric sequences Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges. 47. {(0.7)n}Find the limit of the sequence {an} if 11nan1+1n, for every integer n 1.Compare the growth rates of {n100} and {en/100} as n .Use Theorem 10.6 to evaluate limnn100nn.Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 9. {n3n4+1}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 10. {n123n12+4}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 11. {3n312n3+1}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. {n5+3n10n3+n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. {tan1(10n10n+4)}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. {cot(n2n+2)}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. {1+cos1n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 28. {ln (n3 + 1) ln (3n3 + 10n)}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 29. {ln sin (1/n) + ln n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 14. {k9k2+1}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 23.{4n4+3n8n2+1}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 12. {2en+1en}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 25. {lnn2ln3n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 26. {5(1.01)n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 27. {2n+1 3n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 28. {100(0.003)n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 29. {(0.5)n + 3(0.75)n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 30. {en+nen}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 31. {3n+1+33n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 32. {3n3n+4n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 33. {(n+1)!n!}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 34. {(2n)!n2(2n+2)!}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 15. {tan1 n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 36. {en/102n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 16. n2+1nLimits of sequences Find the limit of the following sequences or determine that the limit does not exist. 18. {n2/n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 19. {(1+2n)n}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 40. {lnnn1.1}Limits of sequences Find the limit of the following sequences or determine that the sequence diverges. 41. {e3n+4n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 20. {(nn+5)n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 21. {(1+12n)n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 22. {(1+4n)3n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 23. {nen+3n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 24. {ln(1/n)n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 25. {(1n)1/n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 26. {(14n)n}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 27. {bn}, where bn={n/(n+1)ifn5000nenifn5000Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 30. {n(1 cos(1/n))}Limits of sequences Find the limit of the following sequences or determine that the limit does not exist. 31. {nsin6n}More sequences Evaluate the limit of the following sequences or state that the limit does not exist. 84. an=n8+n7n7+n8lnnPlot a graph of the sequence {an} for an=sinn2. Then determine the limit of the sequence or explain why the sequence diverges.Plot a graph of the sequence {an} for an=(1)nnn+1. Then determine the limit of the sequence or explain why the sequence diverges.More sequences Find the limit of the following sequences or determine that the sequence diverges. 55. {(1)n2n}More sequences Find the limit of the following sequences or determine that the sequence diverges. 56. {(1)nn}More sequences Find the limit of the following sequences or determine that the sequence diverges. 57. an=(1)nnnSqueeze Theorem Find the limit of the following sequences or state that they diverge. 56. {cos(n/2)n}Squeeze Theorem Find the limit of the following sequences or state that they diverge. 58. {nsin3(n/2)n+1}More sequences Find the limit of the following sequences or determine that the sequence diverges. 60. {(1)n+1n22n3+n}More sequences Find the limit of the following sequences or determine that the sequence diverges. 61. an = en cos nMore sequences Find the limit of the following sequences or determine that the sequence diverges. 62. an=en2sin(en)More sequences Find the limit of the following sequences or determine that the sequence diverges. 63. {tan1nn}More sequences Find the limit of the following sequences or determine that the sequence diverges. 64. {(2n3+n)tan1nn3+n}Squeeze Theorem Find the limit of the following sequences or state that they diverge. 53. {cosnn}Squeeze Theorem Find the limit of the following sequences or state that they diverge. 54. {sin6n5n}Squeeze Theorem Find the limit of the following sequences or state that they diverge. 55. {sinn2n}More sequences Find the limit of the following sequences or determine that the sequence diverges. 68. {1nx2dx}More sequences Find the limit of the following sequences or determine that the sequence diverges. 69. {75n199n+5nsinn8n}More sequences Find the limit of the following sequences or determine that the sequence diverges. 70. {cos(0.99n)+7n+9n63n}Periodic dosing Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume also that aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated. a. Find a recurrence relation for the sequence {dn} that gives the amount of drug in the blood after the nth dose, where d1 = 80. b. Using a calculator, determine the limit of the sequence. In the long run, how much drug is in the persons blood? c. Confirm the result of part (b) by finding the limit of {dn} directly.Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 75. {n!nn}Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 76. {3nn!}Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 77. {n10ln20n}Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 78. {n10ln1000n}Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 79. {n10002n}Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 80. an=4n+5n!n!+2nGrowth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 81. an=6n+3n67+n100Growth rates of sequences Use Theorem 10.6 to find the limit of the following sequences or state that they diverge. 82. an=7nn75nExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If limnan=1 and limnbn=3, then limnbnan=3. b. If limnan=0 and limnbn=, then limnanbn=0. c. The convergent sequences {an} and {bn} differ in their first 100 terms, but an = bn, for n 100. It follows that limnan=limnbn. d. If an={1,12,13,14,15,...} and bn={1,0,12,0,13,0,14,0,...}, then limnan=limnbn. e. If the sequence {an} converges, then the sequence {(1)n an} converges. f. If the sequence {an} diverges, then the sequence {0.000001 an} diverges.Sequences by recurrence relations The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5. a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing. b. Use analytical methods to find the limit of the sequence. an+1=12an+2;a0=1Sequences by recurrence relations The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5. a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing. b. Use analytical methods to find the limit of the sequence. 85. an+1 = 2an(1 an); a0 = 0.3Sequences by recurrence relations The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5. a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing. b. Use analytical methods to find the limit of the sequence. 86. an+1 = 12(an+2an); a0 =2Sequences by recurrence relations The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5. a. Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing. b. Use analytical methods to find the limit of the sequence. 87. an+1 = 2+an; a0 =3Drug Dosing A patient takes 75 mg of a medication every 12 hours; 60% of the medication in the blood is eliminated every 12 hours. a. Let dn equal the amount of medication (in mg) in the bloodstream after n doses, where d1 = 75. Find a recurrence relation for dn. b. Show that {dn} is monotonic and bounded, and therefore converges. c. Find the limit of the sequence. What is the physical meaning of this limit?90E91E92ESuppose the sequence {an}n=0 is defined by the recurrence relation an+1=13an+6; a0 = 3. a. Prove that the sequence is increasing and bounded. b. Explain why {an}n=0 converges and find the limit.Suppose the sequence {an}n0 is defined by the recurrence relation an+1 = an+20 a0 = 6. a. Prove that the sequence is decreasing and bounded. b. Explain why {an}n=0 converges and find the limit.Repeated square roots Consider the sequence defined by an+1 = 2+an, a0 = 2, for n = 0, 1, 2, 3, .... a. Evaluate the first four terms of the sequence {an}. State the exact values first, and then the approximate values. b. Show that the sequence s increasing and bounded. c. Assuming the limit exists use the method of Example 5 to determine the limit exactly.97E98EThe hailstone sequence Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer N and call it a0. This is the seed of a sequence. The rest of the sequence is generated as follows: For n = 0, 1, 2, an+1={an/2ifaniseven3an+1ifanisodd. However, if an = 1 for any n, then the sequence terminates. a. Compute the sequence that results from the seeds N = 2, 3, 4, , 10. You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers N, the sequence terminates after a finite number of terms. b. Now define the hailstone sequence {Hk}, which is the number of terms needed for the sequence {an} to terminate starting with a seed of k. Verify that H2 = 1, H3 = 7, and H4 = 2. c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?Formal proofs of limits Use the formal definition of the limit of a sequence to prove the following limits. 69. limn1n=0102E103E104E105E106EComparing sequences with a parameter For what values of a does the sequence {n!} grow faster than the sequence {nan}? (Hint: Stirlings formula is useful: n!2nnnen, for large values of n.)Reindexing Express each sequence {an}n=1 as an equivalent sequence of the form {bn}n=3. {2n+1}n=1Reindexing Express each sequence {an}n=1 as an equivalent sequence of the form {bn}n=3. {n2+6n9}n=1Prove that if {an} {bn}(as used in Theorem 10.6), then{can} {dbn}, where c and d are positive real numbers.111EWrite the nth partial sum of the series k=0bn.Write the following sums are not geometric sums? a. k=010(12)k b. k=0201k c. k=030(2k+1)3QC4QC5QCExplain why if k=1ak a converges, then the series k=5ak (with a different starting index) also converges. Do the two series have the same value?7QCWhat is meant by the ratio of a geometric series?2EDoes a geometric series always have a finite value?4EFind the first term a and the ratio r of each geometric series. a. k=023(15)k b. k=213(13)kWhat is the condition for convergence of the geometric series k=0ark?Find a formula for the nth partial sum Sn of k1(1k+31k+4). Use your formula to find the sum of the first 36 terms of the series.Reindex the series k=534k263 so that it starts at k = 1.Geometric sums Evaluate each geometric sum. 7. k=083kGeometric sums Evaluate each geometric sum. 8. k=010(14)kGeometric sums Evaluate each geometric sum. 9. k=020(25)2kGeometric sums Evaluate each geometric sum. 10. k=4122kGeometric sums Evaluate each geometric sum. 11. k=09(34)kGeometric sums Evaluate each geometric sum. 13. k=06kGeometric sums Evaluate each geometric sum. 17. 14+112+136+1108++1291616EPeriodic savings Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate d vided by 12 (for example, if tine annual interest rate is 2.4%, r 0.024/12 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m | m(l | r). Continuing in this fashion, it can be shown that tie amount of money in your account after n months is An m + m(1 + r) + + m( 1 + r)n . Use geometric sums to determine the amount of money in your savings account after 5 years (60 months ) using the given monthly deposit and interest rate. 17. Monthly deposits of 250 at a monthly interest rate of 0.2%Evaluating geometric series two ways Evaluate each geometric series two ways. a. Find the nth partial sum Sn of the series and evaluate limnSn. b. Evaluate the series using Theorem 10.7. 18. k0(25)kEvaluating geometric series two ways Evaluate each geometric series two ways. a. Find the nth partial sum Sn of the series and evaluate limnSn. b. Evaluate the series using Theorem 10.7. 19. k=0(27)kEvaluating geometric series two ways Evaluate each geometric series two ways. a. Find the nth partial sum Sn of the series and evaluate limnSn. b. Evaluate the series using Theorem 10.7. 20. 2+43+89+1627+Geometric series Evaluate each geometric series or state that it diverges. 19. k=0(14)kGeometric series Evaluate each geometric series or state that it diverges. 20. k=0(35)kGeometric series with alternating signs Evaluate each geometric series or state that it diverges. 35. k=0(910)kGeometric series with alternating signs Evaluate each geometric series or state that it diverges. 36. k=1(23)kGeometric series Evaluate each geometric series or state that it diverges. 21. k=00.9kGeometric series Evaluate each geometric series or state that it diverges. 22. 1+27+2272+2373+Geometric series Evaluate each geometric series or state that it diverges. 23. 1+1.01+1.012+1.013+Geometric series Evaluate each geometric series or state that it diverges. 24. 1+1+12+13+Geometric series Evaluate each geometric series or state that it diverges. 25. k=1e2kGeometric series Evaluate each geometric series or state that it diverges. 26. m=252mGeometric series Evaluate each geometric series or state that it diverges. 27. k=123kGeometric series Evaluate each geometric series or state that it diverges. 28. k=334k7kGeometric series Evaluate each geometric series or state that it diverges. 29. k=415kGeometric series Evaluate each geometric series or state that it diverges. 30. k=0(43)kGeometric series Evaluate each geometric series or state that it diverges. 35. k=03()kGeometric series with alternating signs Evaluate each geometric series or state that it diverges. 38. k=1(e)kGeometric series Evaluate each geometric series or state that it diverges. 31. 1+e+e22+e33+Geometric series Evaluate each geometric series or state that it diverges. 32. 116+364+9256+271024+Geometric series with alternating signs Evaluate each geometric series or state that it diverges. 39. k=2(0.15)kGeometric series with alternating signs Evaluate each geometric series or state that it diverges. 40. k=13(18)3kGeometric series Evaluate each geometric series or state that it diverges. 41. k=1412kGeometric series Evaluate each geometric series or state that it diverges. 42. k=23ekPeriodic doses Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and he fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount medication in your blood just after your nth dose is An = m + mf + + mfn1. For the given values of :and m, calculate A5, A10, A30 and limnAn. Interpret the meaning of the limit limnAn. 43. = 0.25, m = 200 mg44EDecimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 42. 0.6=0.666Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 41. 0.3=0.333Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 45. 0.09=0.090909Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 47. 0.037=0.037037Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 50. 1.25=1.252525Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 51. 0.456=0.456456456Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 54. 5.1283=5.12838383Decimal expansions Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). 53. 0.00952=0.00952952Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 56. k=1(1k+21k+3)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 55. k=1(1k+11k+2)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 56. k=12025k2+15k4Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 57. k=11(k+6)(k+7)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 58. k=3104k2+32k+63Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 59. k=34(4k3)(4k+1)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 58. k=01(3k+1)(3k+4)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 61. k=1lnk+1kTelescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 60. k=32(2k1)(2k+1)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 63. k=11(k+p)(k+p+1), where p is a positive integerTelescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 64. k=11(ak+1)(ak+a+1), where a is a positive integerTelescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 65. k=1(1k+11k+3)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 66. k=16k2+2kTelescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 67. k=13k2+2k+4Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 68. k=1(k+1k)Telescoping series For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sn}. Then evaluate limnSn to obtain the value of the series or state that the series diverges. 68. k=1(tan1(k+1)tan1k)Evaluating an infinite series two ways Evaluate the series k=1(12k12k+1) two ways. a. Use a telescoping series argument. b. Use a geometric series argument with Theorem 10.8.Evaluating an infinite series two ways Evaluate the series k=1(43k43k+1) two ways. a. Use a telescoping series argument. b. Use a geometric series argument with Theorem 10.8.72EEvaluating series Evaluate each series or state that it diverges. 73. k=1116k2+8k3Evaluating series Evaluate each series or state that it diverges. 74. k=0(sin((k+1)2k+1)sin(k2k1))Evaluating series Evaluate each series or state that it diverges. 75. k=0(14)k53kEvaluating series Evaluate each series or state that it diverges. 76. k=2(38)3kEvaluating series Evaluate each series or state that it diverges. 71. k=1(2)k3k+1Evaluating series Evaluate each series or state that it diverges. 70. k=1(sin1(1/k)sin1(1/(k+1)))Evaluating series Evaluate each series or state that it diverges. 73. k=2ln((k+1)k1)(lnk)ln(k+1)Evaluating series Evaluate each series or state that it diverges. 72. k=1kek+1Evaluating series Evaluate each series or state that it diverges. 81. k=0(3(25)k2(57)k)Evaluating series Evaluate each series or state that it diverges. 82. k=1(2(35)k+3(49)k)Evaluating series Evaluate each series or state that it diverges. 83. k=1(13(56)k+35(79)k)84EEvaluating series Evaluate each series or state that it diverges. 85. k=1((16)k+(13)k)Evaluating series Evaluate each series or state that it diverges. 86. k=123k6kExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. If k=1ak converges, then k=10ak converges. b. If k=1ak diverges, then k=10ak diverges. c. If ak converges, then (ak + 0.0001) converges. d. If pk diverges, then (p + 0.001)k diverges, for a fixed real number p. e. k=1(e)k is a convergent geometric series. f. If the series k=1ak converges and |a| |b|, then the series k=1bk converges. g. Viewed as a function of r, the series 1 + r + r2 + r3 + takes on all values in the interval (12,)Binary numbers Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbersnumbers consisting entirely of 0s and 1s. For this exercise, we consider binary numbers that have the form 0. b1b2b3 where each of the digits b1, b2, b3, ... is either 0 or 1. The base-10 representation of the binary number 0. b1b2b3 is the infinite series b121+b222+b323+. 88. Verify that the base-10 representation of the binary number 0.010101... (which can also be written as 0. 01) is 13.89EFor what value of r does 1 + r + r2 + r3 + = 10?91E92E93ELoans Suppose you borrow P dollars from a bank at a monthly interest rate of r and you make monthly payments of M dollars/month to pay off the loan (for example, if your borrow 30,000 at a rate of 1.5% per month, then P = 30, 000 and r = 0.015). Each month, the bank first adds interest to the loan balance and then subtracts your monthly payment to determine the new balance on your loan. So if A0 is your original loan balance and An equals your loan balance after n payments, then your balance after n payments is given by the recursive formula An = An1(l + r) M, A0 = P. 94. Car loan Suppose you borrow 20,000 for a new car at a monthly interest rate of 0.75%. If you make payments of 600/month, after how many months will the loan balance be zero?95E96EProperty of divergent series Prove Properly 2 of Theorem 10.8: If ak diverges, then cak also diverges for any real number c 0.98E99EDouble glass An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction p of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.Snowflake island fractal The fractal called the snowflake island (or Koch island) is constructed as follows: Let I0 be an equilateral triangle with sides of length 1. The figure I1 is obtained by replacing the middle third of each side of I0 with a new outward equilateral triangle with sides of length 1/3 (see figure). The process is repeated where In+1 is obtained by replacing the middle third of each side of In with a new outward equilateral triangle with sides of length 1/3n + 1. The limiting figure as n is called the snowflake island. a. Let Ln be the perimeter of In. Show that limnLn=. b. Let An be the area of In. Find limnAn. It exists!Remainder term Consider the geometric series S=k=0rk, which has the value 1/(1 r) provided |r| 1. Let Sn=k=0n1rk=1rn1r be the sum of the first n terms. The magnitude of the remainder Rn is the error in approximating S by Sn. Show that Rn=SSn=rn1r.Functions defined as series Suppose a function f is defined by the geometric series f(x)=k=0xk. a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible. b. What is the domain of f?Functions defined as series Suppose a function f is defined by the geometric series f(x)=k=0(1)kxk. a. Evaluate f(0), f(0.2), f(0.5), f(1), and f(1.5), if possible. b. What is the domain of f?110E111EApply the Divergence Test to the geometric series rk. For what values of r does the series diverge?Which of the following series are p-series, and which series converge? a. k=1k0.8 b. k=12k c. k=10k43QCIf we know that limkak=1, then what can we say about k=1ak?Is it true that if the terms of a series of positive terms decrease to zero, then the series converges? Explain using an example.If we know that k=1ak = 10,000, then what can we say about limkak?For what values of p does the series k=11kp converge? For what values of p does it diverge?For what values of p does the series k=101kp converge (initial index is 10)? For what values of p does it diverge?Explain why the sequence of partial sums for a series with positive terms is an increasing sequence.Define the remainder of an infinite series.8EDivergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 9. k=0k2k+1Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 10. k=1kk2+1Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 13. k=011000+kDivergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 14. k=1k3k3+1Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 11. k=2klnkDivergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 12. k=1k22kDivergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 15. k=2kln10kDivergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. 18. k=1k3k!Integral Test Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 23. k=01k+8Integral Test Use the Integral Test to determine whether the fallowing series converge after showing that the conditions of the Integral Test are satisfied. 18. k=11(2k+4)2Integral Test Use the Integral Test to determine whether the fallowing series converge after showing that the conditions of the Integral Test are satisfied. 19. k=115k+33Integral Test Use the Integral Test to determine whether the fallowing series converge after showing that the conditions of the Integral Test are satisfied. 20. k=1ek1+e2kIntegral Test Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 21. k=1ke2k2Integral Test Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 24. k=21k(lnk)2Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series lest to determine whether the following series converge. 23. k=1k1/5Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series lest to determine whether the following series converge. 24. k=11k3Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series lest to determine whether the following series converge. 25. k=1k3ek4Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series lest to determine whether the following series converge. 26. k=1k2k3+5Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series lest to determine whether the following series converge. 27. k=1k1/kDivergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series lest to determine whether the following series converge. 28. k=1k2+1kp-series Determine the convergence or divergence of the following series. 29. k=11k10p-series Determine the convergence or divergence of the following series. 30. k=2kekp-series Determine the convergence or divergence of the following series. 31. k=31(k2)4p-series Determine the convergence or divergence of the following series. 32. k=12k3/2Integral Test Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 25. k=1kekIntegral Test Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 26. k=31k(lnk)lnlnkDivergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge. 35. k=1(kk+10)kIntegral Test Use the Integral Test to determine the convergence or divergence of the following series, or state that the test does not apply. 28. k=1k(k2+1)3p-series Determine the convergence or divergence of the following series. 33. k=11k3p-series Determine the convergence or divergence of the following series. 34. k=1127k23Lower and upper bounds of a series For each convergent series and given value of n, use Theorem 10.13 to complete the following. a. Use Sn to estimate the sum of the series. b. Find an upper bound for the remainder Rn. c. Find lower and upper bounds ( Ln and Un, respectively) for the exact value of the series. 39. k=11k7; n = 240ERemainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than 103. c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. 35. k=11k6Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than 103. c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. 38. k=21k(lnk)2Remainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than 103. c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. 37. k=113kRemainders and estimates Consider the following convergent series. a. Find an upper bound for the remainder in terms of n. b. Find how many terms are needed to ensure that the remainder is less than 103. c. Find lower and upper bounds (Ln and Un, respectively) on the exact value of the series. d. Find an interval in which the value of the series must lie if you approximate it using ten terms of the series. 42. k=1kek2Estimate the series k11k7 to within 104 of its exact value.Estimate the series k11(3k+2)2 to within 103 of its exact value.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
The sum is a p-series.
The sum is a p-series.
Suppose f is a continuous, positive, decreasing function, for x ≥ 1. and ak = f(k), for k = 1, 2, 3, … If converges to L, then converges to L.
Every partial sum Sn of the series underestimates the exact value of .
If ∑k−p converges, then ∑k−p+0.001 converges.
If, then ∑ak converges.
Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 48. k=1ln(2k61+k6)Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 49. k=21ekChoose your test Determine whether the following series converge or diverge. 52. k=1k+1kChoose your test Determine whether the following series converge or diverge. 53. k=11(3k+1)(3k+4)Choose your test Determine whether the following series converge or diverge. 54. k=010k2+9Choose your test Determine whether the following series converge or diverge. 55. k=1kk2+1Choose your test Determine whether the following series converge or diverge. 56. k=12k+3k4kChoose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 55. k=34klnkChoose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 56. k=11k2+7k+12Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 57. k=1(56)kChoose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 58. k=1(k+6k)kChoose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 59. k=43(k3)4Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 60. k=13kk2+1Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 61. 242+252+262+Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 62. 13+15+17+Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4. 63. k=13k+25kLog p-series Consider the series k=21k(lnk)p, where p is a real number. a. Use the Integral Test to determine the values of p for which this series converges. b. Does this series converge faster for p = 2 or p = 3? Explain.Loglog p-series Consider the series k=31k(lnk)(lnlnk)p, where p is a real number. a. For what values of p does this series converge? b. Which of the following series converges faster? Explain. k=21k(lnk)pork=31k(lnk)(lnlnk)p?66EThe zeta function The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by (x)=k=11kx. When x is a real number, the zeta function becomes a p series. For even positive integers p, the value of (p) is known exactly. For example, k=11k2=26,k=11k4=490,andk=11k6=6945,. Use the estimation techniques described in the text to approximate (3) and (5) (whose values are not known exactly) with a remainder less than 103.Reciprocals of odd squares Assume that k=11k2=26 (Exercises 65 and 66) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers.Gabriels wedding cake Consider a wedding cake of infinite height, each layer of which is a right circular cylinder of height 1. The bottom layer of the cake has a radius of 1, the second layer has a radius of 1/2. the third layer has a radius of 1/3, and the nth layer has a radius of 1/n (see figure). a. To determine how much frosting is needed to cover the cake, find the area of the lateral (vertical) sides of the wedding cake. What is the area of the horizontal surfaces of the cake? b. Determine the volume of the cake. (Hint: Use the result of Exercise 66.) c. Comment on your answers to parts (a) and (b).