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## Calculus Several Variables Study Guide

by: Avid Notetaker

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# Calculus Several Variables Study Guide Math 010A

Marketplace > University of California Riverside > Mathematics (M) > Math 010A > Calculus Several Variables Study Guide
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COURSE
Calculus:Several Variables
PROF.
Meng Zhu
TYPE
Study Guide
PAGES
1
WORDS
CONCEPTS
Calculus, Series, study, guide
KARMA
50 ?

## Popular in Mathematics (M)

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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