Calculus Several Variables Study Guide
Calculus Several Variables Study Guide Math 010A
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This 1 page Study Guide was uploaded by Avid Notetaker on Thursday April 21, 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 24 views. For similar materials see Calculus:Several Variables in Mathematics (M) at University of California Riverside.
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Date Created: 04/21/16
NAME CONVERGENCE TESTS FOR INFINITE SERIES COMMENTS STATEMENT Geometric series ! ar k = 1 , if –1 < r < 1 Geometric series converges if and diverges otherwise –1 < r < 1 Divergence test – r If p If lim a k= 0, !a kay or may not converge. (nth Term test) lim ak " 0, then !a kiverges. ! k ! Integr If p is a real constant, the series ! 1 = + + . . . + + . . . – series a p p p p converges if p > 1 and diverges if 0 < p # 1. !a has positive terms, let f(x) b as tnhcitens aht ns fu(lis heeans yk tios integrate. This k e p$l a beyn in the formula for u . kIf is decreasing and continuous for !a 1% Comparison test (Direct) test only applies to series with positive terms. bok acdn vfe(g)e doxr bo th diverge. If !a k and !b k are series itshehsisivset ta s ssute hratt ache trm stnr !eak ften i le) sihae "itisg cgesrpes"d i!nbg term in !bk, then series" !a k converges, then the "smaller easier to apply. This test only applies to series Limit Comparison test (b) if the "smkller verieess". !a with positive terms. series" !b k diverges, then the "bigger diverges. k If !a k and !b kre series with positive terms su c h Tthhaits is easier to apply than the comparison test, if L > 0, =e n then both series converge or both diverge. k ! b k but still requires some skill in choosing the Ratio test if L = 0, and series !b kor comparison. if L = +% a!nbdk! bconverges, then !a konverges. k diverges, then !a dkverges. If !a k is a series with positive term s srch t hti st when a lim ak+1 powers. k involves factorials or k th Root test k he! ifaL < 1 t ,e series converges k if L > 1 or L = +%, the series diverges if L = 1, another test must be used. If !a k is a series with positive term s ucry hist test when a if L < 1a te s eries (a )erge s= L, then ! k k ! k Alternating Series test k involves k th powers. if L > 1 or L = +%, the series diverges if L = 1, another test must be used. The series a Alternating Series Estimation Theor eIm f t:he alternating series ! ( conv er agif a – a + . . . and –a + a – a + a – . . . k+1 1 2 3 4 1 2 3 4 converges, then the truncat– io1n) errk ar for the n partial sum is less than a th (1) a if an alternating series cn+1 , e e.s, then the error (Leibniz's Theorem) The 1eri e2s d3i vaer > . . . and (2) ak = 0 ! in estimating the sum using Absolute Convergence and terms is less than the n+1 term. ges if lim ak " 0 k ! If !a ik ! |ias a series with nzeortoe tt sih asrioe ceogese,tes na:bsolutely, then it if |aerges, i.e. | converges, then !a converges. if !ka | converges, then !a kconverges absolutely. k k Conditional Convergence Otherwikse,i arges, then !a konverges conditionally. diverges. k
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