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162 Exam III Study Guide

# 162 Exam III Study Guide - Fall 2011 MA 16200 Study Guide...

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Fall 2011 MA 16200 Study Guide - Exam # 3 (1) Sequences; limits of sequences; Limit Laws for Sequences; Squeeze Theorem; monotone sequences; bounded sequences; Monotone Sequence Theorem. (2) Infinite series n =1 a n ; sequence of partial sums s n = n k =1 a k ; the series n =1 a n converges to s if s n s . (3) Special Series : (a) Geometric Series : n =1 ar n - 1 = a (1 + r + r 2 + r 3 + · · · ) = a 1 r , if | r | < 1 (converges). The Geometric Series diverges if | r | ≥ 1. (b) p-Series : n =1 1 n p converges when p > 1; diverges when p 1. (4) List of Convergence Tests for n =1 a n . (0) Divergence Test (1) Integral Test (2) Comparison Test (3) Limit Comparison Test (4) Alternating Series Test (5) Ratio Test (6) Root Test (Useful inequality: ln x < x m , for any m > 1 e > 0 . 37.) (5) Strategy for Convergence/Divergence of Infinite Series . Usually consider the form of the series a n : (i) If lim n →∞ a n ̸ = 0 or DNE, the use Divergence Test . (ii) If series is a p -Series 1 n p , use p-Series conclusions; if series is a Geometric series a r n - 1 , use Geometric Series conclusions.

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