CONVERGENCE TESTS FOR INFINITE SERIES NAME COMMENTS STATEMENT Geometric series ! ar k = a 1 – r , if –1 < r < 1 Geometric series converges if –1 < r < 1 and diverges otherwise Divergence test (nth Term test) If k ! ! lim a k " 0, then ! a k diverges. If k ! ! lim a k = 0, ! a k may or may not converge. p – series If p is a real constant, the series ! 1 a p = 1 1 p + 1 2 p + . . . + 1 n p + . . . converges if p > 1 and diverges if 0 < p # 1. Integral test ! a k has positive terms, let f(x) be a function that results when k is replace by x in the formula for u k . If is decreasing and continuous for x $ 1, then ! a k and ! " 1 % f(x) dx both converge or both diverge. Use this test when f(x) is easy to integrate. This test only applies to series with positive terms. Comparison test (Direct) If ! a k and ! b k are series with positive terms such that each term in ! a k is less than its corresponding term in ! b k , then (a) if the "bigger series" ! b k converges, then the "smaller
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This note was uploaded on 02/22/2012 for the course MATH 9c taught by Professor C during the Fall '11 term at UC Riverside.