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This sheet gives a list of tests used for determining whether a series converges or diverges.
Calculus:Several Variables
Meng Zhu
Study Guide
Calculus, Divergence, Convergence, Tests, Math, Limits
50 ?




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This 30 page Study Guide was uploaded by Avid Notetaker on Monday February 22, 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 72 views. For similar materials see Calculus:Several Variables in Mathematics (M) at University of California Riverside.

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Popular in Mathematics (M)




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Date Created: 02/22/16
CONVERGENCE TESTS FOR INFINITE SERIES NAME COMMENTS STATEMENT k ! ar = a , if –1 < r < 1 Geometric series converges if –1 < r < 1 Geometric series 1 – r and diverges otherwise lim Divergence test If lim a " 0, then !a diverges. If      a  = 0, !a  may or may not converge. k  k k (nth Term test) k  ! k k If p is a real constant, the series ! 1 = 1 + 1 + . . . + 1 + . . . p – series p p p p a 1 2 n converges if p > 1 and diverges if 0 < p # 1. !ak has positive terms, let f(x) be a function that results when k is replace by x in the formula fok u . If is decreasing and continuous for Integral test x $ 1, then Use this test when f(x) is easy to integrate. This % !ak and  test only applies to series with positive terms. f(x) dx 1 both converge or both diverge. If !k  and !k  are series with positive terms such that each term ink!a is less than its corresponding term in !bk, then Comparison test (Direct) (a) if the "bigger series" kb  converges, then the "smaller Use this test as a last resort. Other test are often easier to apply. This test only applies to series series" !ak converges. with positive terms. (b) if the "smaller series" !a diverges, then the "bigger k series" !bk diverges. If !k  and !bk are series with positive terms such that lim a k = L k  ! bk 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 !k  for comparison. if L = 0, and !k  converges, then !a kconverges. if L = +% and !b kdiverges, then !a kdiverges. If !k  is a series with positive terms such that Ratio test lim a k+1 = L , Try this test when akinvolves factorials or kh powers. k  ! 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 !k  is a series with positive terms such that lim = lim  1/k ) = L, then ka k (a k Root test k  ! k  ! Try this test when a kinvolves k t powers. 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 2 – a + a – . . . k+1 If the alternating series ! (–1)  a 1 2 3 4 1 3 4 k converge if th converges, then the truncation error for the n Alternating Series test partial sum is less than a n+1 , i.e. (Leibniz's Theorem) lim if an alternating series converges, then the error (1)  a  > a  > a  > . . . and 1 2 3 (2) k  !k = 0 in estimating the sum using st n terms is less than the n+1  term. lim The series diverges if ak k  ! " 0 If !k  is a series with nonzero terms that converges, then: Note that if a series converges absolutely, then it converges, i.e. Absolute Convergence and if !|a | converges, then !a k converges absolutely. if !|a | converges, then !a  converges. k k Conditional Convergence k if !|k | diverges, then !ak converges conditionally. Otherwise, !ak diverges.


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