quiz05_s107_solns - for just yes or no; do not refer to...

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

View Full Document Right Arrow Icon
Math 1B Quiz 5 Section 107 October 13, 2009 1. (5 points) Say whether the following series converge and clearly explain why. (A “yes” or “no” without explanation is worth 0 points, even if correct.) (a) X n =2 1 n 5 / 4 ln n Since 1 n 5 / 4 ln n < 1 n 5 / 4 for all n 2, n =2 1 n 5 / 4 ln n converges if n =2 1 n 5 / 4 does. But n =2 1 n 5 / 4 is a p -series with p > 1, so it converges. Therefore n =2 1 n 5 / 4 ln n converges. (b) X n =2 1 n ln n For n 2, x and ln x are positive and increasing, so 1 x ln x is positive, decreasing, and continuous. Thus by the integral comparison test, n =2 1 n ln n converges if and only if R 2 1 x ln x dx does. But letting u = ln x , we have R 2 1 x ln x dx = R ln 2 1 u du , which diverges. Therefore, n =2 1 n ln n diverges. (c) X n =2 1 n 4 / 5 ln n Because n 4 / 5 < n for all n 2, we have that 1 n 4 / 5 ln n > 1 n ln n for all n 2. By the comparison test, since n =2 1 n ln n diverges, n =2 1 n 4 / 5 ln n diverges also.
Background image of page 1

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

View Full DocumentRight Arrow Icon
2. (3 points) Does the following series converge? Clearly explain your answer (no points
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: for just yes or no; do not refer to behavior). X n =1 n 2 n 6 + 1 I will use the Limit Comparison Test to compare n 2 n 6 +1 to 1 n . (The inspiration to choose 1 n was by thinking about the behavior of n 2 n 6 +1 as n gets large, but talk about approximating behaviors does not count as a solution for this class.) So I consider lim n 1 n n 2 n 6 +1 = lim n n 6 + 1 n 3 = p 1 + 1 /n 6 1 = 1 . By the Limit Comparison Test, since n =1 1 n diverges, so does n =1 n 2 n 6 +1 . 3. (2 points) Give an example of (a) an alternating series that converges n =1 (-1) n n (b) an alternating series that does not converge n =1 (-1) n...
View Full Document

Page1 / 2

quiz05_s107_solns - for just yes or no; do not refer to...

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