mth.122.handout.16

mth.122.handout.16 - For what values of p does the series...

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MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #16 — Sakai Web Project Material 1 Introduction to Sequences and Series, Part III 1. Convergence Properties of Series (a) If X n =1 a n and X n =1 b n converge, and if k is a constant, then i. X n =1 ( a n + b n ) converges to X n =1 a n + X n =1 b n . ii. X n =1 ka n converges to k X n =1 a n . (b) Changing a finite number of terms in a series does not change whether or not it converges, although it may change the value of its sum if it does not converge. (c) If lim n →∞ a n 6 = 0, or lim n →∞ a n = DNE, then X n =1 a n is divergent. (d) If X n =1 a n diverges, then X n =1 ka n diverges if k 6 = 0. 2. The Integral Test : Suppose f is a continuous, positive, decreasing function on [1 , ) and a n = f ( n ). Then the series X n =1 a n is convergent if and only if the improper integral Z 1 f ( x ) d x is convergent. In other words: (a) If Z 1 f ( x ) d x is convergent, then X n =1 a n is convergent. (b) If Z 1 f ( x ) d x is divergent, then X n =1 a n is divergent. Example:
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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 n-1) 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 x-1 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 ( n-1) ( 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 to-ln 2 . 6 Diverges. 4...
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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.

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mth.122.handout.16 - For what values of p does the series...

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