' (x-1)(x +1)Σ-υ2. For the seriesn=1пi) use the Ratio Test to determine for what values of x does this series convergeii) for what values of x does the above series diverge.iii) Use the alternating series test to determine the convergence or divergence ofΣ-υ1n=1n+1iv) Determine the convergence or divergence of . Justify your answer.n=14 m3.

Answered: Image /qna-images/answer/aa6ad021-cbae-4751-aba8-b53796c17133.jpg

Answered: ' (x-1)(x +1) Σ-υ 2. For the series n=1…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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V12(-1)" 3" (2n + 1)" 1. Consider the series n=0 (a) Use any test for convergence/divergence to show that the series converges. V12(-1)" 3" (2n + 1) (b) It is possible to show that the sum of the series is T, in other words, the series n=0 converges to the number T. (You do NOT need to prove this, but it can be done somewhat easily using a Taylor series expansion of arctan x.) Suppose you want to use a partial sum of this series to estimate the value of T to an accuracy of within 0.0001. Would using the first 8 terms of the series be enough to ensure you get an accuracy of within 0.0001? (8 terms means the terms where n = 0, 1, 2, 3, ..., 7.) Hint: Use Theorem 5.14.
Problem 6. ∞ The series Σ diverges. n=2 1 n+1 (a) You cannot show divergence for this series using NTDT. Why? (b) You cannot show divergence for this series using the Comparison Test with n=1 1 n Why?
A series of the form 00 2(-1)" a, = a, – az + a, – a̟ +..+(-1)"-' a, + ·…. n=1 converges if 0
1 Expand the function 4+5x in a power series E anx" with center c = 0. Find a,x". n=0 (Express numbers in exact form. Use symbolic notation and fractions where needed. For alternating series, include a factor of the form (-1)" in your answer.) anx" DNE Incorrect Determine the interval of convergence. (Give your answers as intervals in the form (*, *). Use symbol ∞ for infinity, u for combining intervals, and appropriate type of parenthesis " (", ") " [" or "] " depending on whether the interval is open or closed. Enter DNE if interval is empty. Express numbers in exact form. Use symbolic notation and fractions where needed.) x E DNE Incorrect
Σαπ M n=1 an = Σ n=1 3n+n 6"+In(n) We want to use the limit comparison test to determine whether this series converges or diverges. (a) The dominant term of the numerator is The dominant term of the denominator is We should therefore use b = We get an L= lim n+oo bn == (b) Which of the following statement is true for the series b₁₁? n=1 A. The series diverges by the nth term test for convergence. OB. The series converges by the geometric series test since -1 1. E. The series diverges by the geometric series test since r = F. none of the above >1 (c) What does the limit comparison test tell you about the series A. The test tells us no information OB. The series diverges C. The series converges D. none of the above +∞0 Σ 3" + n¹ 6n+ln(n) ? n=1
6
(1-2)". Let us apply the root test on the series Setting a, = we have + n=1 L = lim Va, = lim n- 00 n- 00 Since L? v 1, we conclude that the series is ?
q5
an where n=1 Consider the series 57+2 an (Зп? + 7)2" In this problem you must attempt to use the Ratio Test to decide whether the series converges. Compute аn+1 L = lim An n00 Enter the numerical value of the limit L if the limit exists, INF if it diverges to infinity, MINF if it diverges to negative infinity, or DIV if it diverges but not to infinity or negative infinity. L = Which of the following statements is true? A. The Ratio Test says that the series converges absolutely. B. The Ratio Test says that the series diverges. C. The Ratio Test says that the series converges conditionally. D. The Ratio Test is inconclusive, but the series converges absolutely by another test or tests. E. The Ratio Test is inconclusive, but the series diverges by another test or tests. F. The Ratio Test is inconclusive, but the series converges conditionally by another test or tests. Enter the letter for your choice here: ? v
(-1)"-1 6. (a) Show that the series is convergent. n2 n=1 (-1)"-1 (b) Find the partial sum s, of the series Estimate the error in using s; as an n2 n=1 approximation to the sum of the series. (c) Find a value of n so that s, is within 0.001 of the sum.
(1+3i)" Determine whether the series converges, converges 23п+1 n=1 absolutely, or diverges. If the series converges, find its sum.

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