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mth.122.handout.17 - MTH 122 Calculus II Essex County...

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MTH 122 — Calculus II Essex County College — Division of Mathematics and Physics 1 Lecture Notes #17 — Sakai Web Project Material 1 Introduction to Sequences and Series, Part IV 1. Comparison Test : Suppose that X n =1 a n and X n =1 b n are series with positive terms. (a) If X n =1 b n is convergent, and 0 < a n b n for all n , then X n =1 a n is also convergent. (b) If X n =1 b n is divergent, and a n b n > 0 for all n , then X n =1 a n is also divergent. Example: Use the comparison test to determine whether X n =1 1 n 3 + 1 converges. Work: For n 1 we know that n 3 n 3 + 1, so 0 < 1 n 3 + 1 1 n 3 . You should note that X n =1 1 n 3 is a convergent p -series. The conclusion, using the Comparison Test , is that X n =1 1 n 3 + 1 also converges . The two big series that you should use for comparisons are the p -series and the geometric series. The harmonic series is a p -series, with p = 1. The Geometric Series X n =1 ar n - 1 = a + ar + ar 2 + · · · is convergent if | r | < 1, and its sum is X n
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