This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: For what values of p does the series ∞ X n =1 1 n p converge? 1 This document was prepared by Ron Bannon ( [email protected] ) using L A T E X2 ε . Last revised January 10, 2009. 1 Work: First just look at the limit lim n →∞ 1 n p for p ≤ 0. Now use the integral test on 0 < p ≤ 1 and p > 1. 2 1.1 Examples 1. Do the series converge or diverge? (a) 2 ∞ X n =1 3 (2 n1) 2 (b) 3 ∞ X n =1 n n + 1 (c) 4 ∞ X n =1 3 n 2 + 4 (d) 5 ∞ X n =1 n + 2 n n 2 n 2 Converges. 3 Diverges. 4 Converges. 5 Diverges. Hint: Z x + 2 x x 2 x d x = ln x1 2 x ln 2 + C 3 (e) 6 ∞ X n =1 n + 1 n 2 + 2 n + 2 2. Consider the series ∞ X n =2 ln ( n1) ( n + 1) n 2 . (a) Show that S 4 is ln (5 / 8). (b) Show that S n is ln n + 1 2 n . (c) Show that this series converges toln 2 . 6 Diverges. 4...
View
Full
Document
This note was uploaded on 01/31/2011 for the course MTH 222 taught by Professor Ban during the Spring '10 term at Essex County College.
 Spring '10
 Ban
 Calculus, Division, Sequences And Series

Click to edit the document details