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# Convergence Tests List Math 010A

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This 1 page Study Guide was uploaded by Avid Notetaker on Monday April 4, 2016. The Study Guide belongs to Math 010A at University of California Riverside taught by Meng Zhu in Winter 2016. Since its upload, it has received 86 views. For similar materials see Calculus:Several Variables in Mathematics (M) at University of California Riverside.

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Date Created: 04/04/16

CONVERGENCE TESTS FOR INFINITE SERIES NAME COMMENTS STATEMENT a Geometric series converges if –1 < r < 1 Geometric series ! ar = 1 – r , if –1 < r < 1 and diverges otherwise lim Divergence test lim If ak= 0, !a kay or may not converge. (nth Term test) If ak" 0, then !a kiverges. k ! ! k ! ! 1 1 1 1 If p is a real constant, the seriesp!= p + p + . . . +p + . . . p – series a 1 2 n converges if p > 1 and diverges if 0 < p # 1. !a kas positive terms, let f(x) be a function that results when k is replace by x in the formula for k . If is decreasing and continuous for x $ 1, then Integral test % Use this test when f(x) is easy to integrate. This test only applies to series with positive terms. !a knd f(!) dx 1 both converge or both diverge. If !akand !b ake series with positive terms such that each term in !a k is less than its corresponding term in !k , then Comparison test (Direct) Use this test as a last resort. Other test are often (a) if the "bigger series" !k converges, then the "smaller series" !akconverges. easier to apply. This test only applies to series with positive terms. (b) if the "smaller series" !k diverges, then the "bigger series" !b kiverges. If !a and !b are series with positive terms such that k k a lim k= L k ! ! b k This is easier to apply than the comparison test, Limit Comparison test but still requires some skill in choosing the if L > 0, then then both series converge or both diverge. series !b kor comparison. if L = 0, and !bkconverges, then !a cknverges. if L = +% and !b dkverges, then !a dkverges. If !akis a series with positive terms such that Ratio test lim ak+1 Try this test when akinvolves factorials or th = L , powers. k ! ! a k then if L < 1, the series converges if L > 1 or L = +%, the series diverges if L = 1, another test must be used. If !a is a series with positive terms such that k lim lim kak = (ak)1/k= L, then Root test k ! ! k ! ! Try this test when a involves k thpowers. k if L < 1, the series converges if L > 1 or L = +%, the series diverges if L = 1, another test must be used. The series Alternating Series Estimation Theorem: a – a + a – a + . . . and –a + a – a + a – . . . k+1 1 2 3 4 1 2 3 4 If the alternating series ! (–1) ak converges, then the truncation error for the n converge if partial sum is less than n+1, i.e. Alternating Series test (Leibniz's Theorem) (1) a > a > a > . . . and (2)lim a = 0 if an alternating series converges, then the error 1 2 3 k ! ! k in estimating the sum using st n terms is less than the n+1 term. The series diverges if lim a " 0 k ! ! k If !a is a series with nonzero terms that converges, then: Note that if a series converges absolutely, then it k converges, i.e. Absolute Convergence and if !|k | converges, then !k converges absolutely. if !|ak| converges, then !akconverges. Conditional Convergence if !|a | diverges, then !a converges conditionally. k k Otherwise, !a dkverges.

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