{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

quiz05_s103_solns

quiz05_s103_solns - for just “yes” or “no” do not...

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

Math 1B Quiz 5 Section 103 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 6 / 5 ln n Since 1 n 6 / 5 ln n < 1 n 6 / 5 for all n 2, n =2 1 n 6 / 5 ln n converges if n =2 1 n 6 / 5 does. But n =2 1 n 6 / 5 is a p -series with p > 1, so it converges. Therefore n =2 1 n 6 / 5 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 5 / 6 ln n Because n 5 / 6 < n for all n 2, we have that 1 n 5 / 6 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 5 / 6 ln n diverges also.

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

View Full Document
2. (3 points) Does the following series converge? Clearly explain your answer (no points
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 √ n 4 + 1 I will use the Limit Comparison Test to compare n √ n 4 +1 to 1 n . (The inspiration to choose 1 n was by thinking about the behavior of n √ n 4 +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 √ n 4 +1 = lim n →∞ √ n 4 + 1 n 2 = lim n →∞ p 1 + 1 /n 4 1 = 1 . By the Limit Comparison Test, since ∑ ∞ n =1 1 n diverges, so does ∑ ∞ n =1 n √ n 4 +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

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern