sg3-f12 - Fall 2012 MA 16200 Study Guide Exam 3(1 Sequences limits of sequences Limit Laws for Sequences Squeeze Theorem monotone sequences bounded

sg3-f12 - Fall 2012 MA 16200 Study Guide Exam 3(1 Sequences...

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Fall 2012 MA 16200 Study Guide - Exam # 3 (1) Sequences; limits of sequences; Limit Laws for Sequences; Squeeze Theorem; monotone se- quences; bounded sequences; Monotone Sequence Theorem. (2) Infinite series X n =1 a n ; sequence of partial sums s n = n X k =1 a k ; the series X n =1 a n converges to s if its sequence of partials sums s n s . (3) GEOMETRIC SERIES : X n =1 ar n - 1 = a + ar + ar 2 + ar 3 + · · · = a (1 + r + r 2 + r 3 + · · · ) = a 1 - r , if | r | < 1. The Geometric Series diverges if | r | ≥ 1. (4) p - SERIES : X n =1 1 n p converges when p > 1; diverges when p 1. (5) HARMONIC SERIES : X n =1 1 n diverges. (6) List of Convergence Tests for X 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 (A useful inequality: ln x < x α , for any fixed constant α 1 2 .) (7) Strategy for Convergence/Divergence of Infinite Series : Usually first look at the form of the series X a n : (i) If lim n →∞ a n 6

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