Limits Evaluate the following limits using Taylor series.
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- The integral tests says that if an=f(n), then the series 2 an is convergent if and only n =1 if the integral J F(x)dx is convergent as long as the function f is BLANK-1, BLANK- 2, and BLANK-3 on the interval X21. BLANK-1 Add your answer BLANK-2 Add your answer BLANK-3 Add your answer .T dx= lim x-2dx= lim -Tl+1¬1= lim +1 = 1 Since the integral converges and therefore the series 2 K=1 K? also converges, and <1+1=2. K=1 K2arrow_forward1 Calculate the difference quotient for f(x) at a = 4. %3D V2x + 9 (Express numbers in exact form. Use symbolic notation and fractions where needed.) f(4 + h) – f(4) h Calculate f'(4) by finding the following limit. f(4 + h) – f(4) lim h→0 h (Give an exact answer. Use symbolic notation and fractions where needed.) f'(4) = Find an equation of the tangent line to f(x) at a = 4. (Express numbers in exact form. Use symbolic notation and fractions where needed.) y =arrow_forwardWithout uding L hospitals rules or series expansion evaluate the following limitarrow_forward
- n 2 3 = 2²2 2 [³ ( 5 + + ² ) ² - 6 (5 + + ²) ²]. i- i n n integral, that is provide a, b and f(x) in the expression Given Rn a = b = f(x) = , express the limit as n → ∞ as a definite Sºf(x)dx.arrow_forwardApproximate the area under the graph of the function f(x) = 6x + 8 from 3 to 7 for n = 4 and n = 8 subintervals by using lower and upper sums. (Use symbolic notation and fractions where needed.) (a) By using lower sums s, (rectangles that lie below the graph of f(x)). lower sum S4 = lower sum S8 = (b) By using upper sums S, (rectangles that lie above the graph of f(x)). upper sum S4 =arrow_forward11. TRUE or FALSE. Identify if the statement is true or false. If True, then prove or explain. If it is False, then explain why or give a counterexample and explain why it works: "If f is continuous on [1,00) and limx→ f(x) = 2 9 then (x) dx converges." 1arrow_forward
- If a series of positive terms converges, does it follow that the remainder R,, must decrease to zero as n-co? Explain. Choose the correct answer below. OA. R, must decrease to zero because lim R, lim f(x)dx for all positive functions x. n-00 71-400 00 lima, n+00 K=1 OC. R, does not decrease to zero because R, is positive for a series with positive terms. OD. R, does not decrease to zero because convergent series do not have remainders. OB. R, must decrease to zero because lim R, n-+00 -0.arrow_forwardlim (n22n) / n! = ?arrow_forward1. (a) Use Bernoulli's inequality to prove that (1+ (b) Prove that Vn √n for all n € N.arrow_forward
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