mt2-ans - Math 1B. Sample Answers to Second Midterm 1. (12...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 1B. Sample Answers to Second Midterm 1. (12 points) Determine whether the series X n =1 (- 1) n n + 1 n 2 + 1 is absolutely convergent, conditionally convergent, or divergent. Following Example 2 on pages 711712, we note that the series is alternating, but lim n n + 1 n 2 + 1 = lim n 1 + 1 /n p 1 + 1 /n 2 = 1 , so the Alternating Series Test cannot be applied. Instead, we look at the limit of the n th term of the series: lim n (- 1) n n + 1 n 2 + 1 . This limit does not exist (the terms alternate between being close to 1 and- 1), so the series diverges, by the Test for Divergence. 2. (14 points) Determine whether the series X n =2 cos n n 2- 1 is absolutely convergent, conditionally convergent, or divergent. Since- 1 cos n 1, we have cos n n 2- 1 1 n 2- 1 . Therefore, we can test for absolute convergence by the Comparison Test, comparing it with the series 1 / ( n 2- 1) ( provided the series 1 / ( n 2- 1) converges). This latter series has positive terms, so we can use the Limit Comparison Test to compare it with the p-series 1 /n 2 : lim n 1 n 2- 1 1 n 2 = lim n 1 1- 1 /n 2 = 1 . Since the p-series converges ( p = 2 > 1), the series 1 / ( n 2- 1) also converges (by the Limit Comparison Test), and so the original series converges absolutely (by the ordinary Comparison Test)....
View Full Document

This note was uploaded on 03/25/2011 for the course MATH 1B taught by Professor Reshetiken during the Fall '08 term at University of California, Berkeley.

Page1 / 4

mt2-ans - Math 1B. Sample Answers to Second Midterm 1. (12...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online