Problem #8: If the sums of the series 3(-1)" (n+2)³ Problem #8: Σ is approximated by 3(-1)" Σ (n+2)³ n-0 Sm find the smallest possible m such that the alternating series error estimate says|s-5m| < 0.03. Save

Problem 3. Define the integer sequence Lo, L1, L2, , recursively by Lo = 2, L1 = 1, and %3D •... Ln+2 = Ln+1 + Ln , for n > 0. Prove that L + L + …+ L = L„Ln+1– 2, for n > 1. ...

(b) (Using the method of trying a solution of the form f(n) = Kwn , etc.,...) find the direct prescription for the following recursively-prescribed sequence f(n + 2) − 5f(n + 1) + 6f(n) = 0, where n is an integer ≥ 1, f(1) = 0, and f(2) = 6.

(In n)ª Problem 2. Use the Alternating Series Test to show that for any real number a, the series >(-–1)"+1 n n=1 converges. (Hint: You need to show that the sequence {(ln n)ª/n} converges to 0 for any a, and is eventually decreasing. Show that the function f(x) = (In x)ª /x is decreasing if x is sufficiently large.)

Problem 4. Does the series n=2 1 √n - 1 converge or diverge? Show all work.

Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent to 0, then the series > (-1)"+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 1 1 1 1 1 1 1 1 (-1)"+1bn, - 2 4 3 8. 4 16 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 4' 3'8'4'16’5'32' 1 1 } n 2n (a) Does this sequence {bn} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 (b) Show that this series diverges. (Hint: This series can also be written as Suppose that it did n 2n n=1 should converge. If you add a certain geometric series to it, you're adding two convergent series together, so you get another convergent series but do you?) (c) Is the Alternating Series Test wrong? Explain why not.

Y.7.

For problem 2 part 2: To get your comparison series for LCT, choose the series with leading term over leading term. So you want the series of sqrt{n} over n  "leading term over leading term" then go from there

3.1.3. Intro To Real Analysis

Problem 3: Life is better when there are options. The series can be analyzed in many п(п - 1) n=3 different ways. You will discover them below! (a) Show this series a telescoping series. (You may need to find a function f(x) so that Sp = f(n), where {Sm} is the associated sequence of partial sums, see the hint in the next part.) Find the sum of this telescoping series. (b) Use the Integral Test to determine whether the series converges or diverges. What is the value of the associated improper integral? Be sure to carefully write out your argument. (Hint: write out the telescoping series in the form E3 A - for some constants A, B and apply the integral test to this series.) (c) Does the sum of the series in part (a) match the value of the improper integral from part (b)? Should it? Briefly explain the reasoning that allowed you to make your conclusion. (A picture might be helpful too!) (d) Extra Credit: Use the Limit Comparison Test to determine whether the series converges or diverges.…