Convergence. and x a R. 2. The series converges at x a. and diverges elsewhere 3. The series converges for every x
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1 Covergece Whe we exmied ifiite series it ws stressed the importce of covergece There re vriety of test tht mkes it esy to determie if ifiite series coverges or diverges The Covergece Theorem for Power Series k There re three possibilities for c x covergece: k 0 k with respect to There exists positive umber R such tht the series for x R but coverges for x R The series my or my ot coverge t either of the edpoits x R d x R 2 The series coverges t x d diverges elsewhere 3 The series coverges for every x The umber R is the rdius of covergece, d the set of ll vlues of x for which the series coverges is the itervl of covergece The most obvious requiremet for covergece of series is tht the th term must go to zero s The th Test The th -Term Test for Covergece diverges if lim fils to exist or is differet from zero if lim 0, the the series diverges 2 WARNING: If lim 0, the series does ot ecessrily coverge The series my coverge or diverge 3 if the series does coverge, the lim 0
2 Exmple Determie the covergece or divergece of 3! 2! 5 Solutio: We hve the 3! 3 lim, sice lim 0 therefore the series diverges 2! 5 2 The Geometric Series The Geometric Series r 0 coverges if r d hs the sum of r 2 diverges if r Exmple Determie the covergece or divergece of 7 5 Solutio: This is geometric series with 7 5 d r 3 Sice r, the series coverges The p-series The p-series p Coverges if p Diverges if p
3 Exmple Determie the covergece or divergece of 7 5 Solutio: 7 5 This is p-series with p Sice p<, the series diverges The Hrmoic Series The Hrmoic Series diverges Note: the Hrmoic Series is just p-series with p= Exmple Determie the covergece or divergece of 7 Solutio: Sice 7 7,which is 7 times diverget series which diverges The Altertig Series The Altertig Series or 0 (series is o-icresig) 2 lim 0 coverges if:
4 Exmple Determie the covergece or divergece of 7 Solutio: Sice which is ltertig series with Now we must check both coditios Is : Yes does lim 0 lim 0 Yes 7 Therefore, the series coverges The Rtio Test The Rtio Test This test will help you determie whether series coverges (We lredy hve test for Geometric Series) Let be series, d suppose tht lim L If L<, the the series coverges bsolutely 2 If L> (or L ifiite), the the series diverges 3 If L=, the test is icoclusive must try other test The Rtio Test is frequetly used o the AP exm
5 Exmple Determie the rdius d itervl of covergece for the power series x 0! Solutio: To determie covergece, we must pply the rtio test x! x L lim lim 0 x! Becuse L=0 for ll x, the series coverses bsolutely for ll x Therefore the rdius of covergece is R d the itervl of covergece is, Exmple Determie the covergece or divergece of 4 Solutio: With 4, we will try the rtio test L 4 4 lim lim lim 4 4 With L= the Rtio Test is icoclusive However, this is p-series with p=4 which is >, thus this series coverges bsolutely
6 The Root Test The Root Test For the series fid lim L if L< the the series coverges bsolutely 2 if L> (or L is ifiite), the the series diverges 3 if L=, the test is icoclusive d other test my be used The root test is rrely used o the AP exm Exmple Determie the covergece or divergece of 3 3 l Solutio: lim lim lim lim 0 l l l Sice 0 is less th, the give series coverses bsolutely
7 The Itegrl Test The Root Test If f is positive, cotiuous, d decresig for x d f, the d f x dx if the improper itegrl f x dx exist, the the series coverges 2 if the improper itegrl f x dx, the the series diverges Either both coverge or both diverge I other words, if you wt to test covergece for series with positive terms, evlute the itegrl d see if the itegrl coverges Exmple Determie the covergece or divergece of 5 3 Solutio: Let f x x c, ote tht f(x) is positive, cotiuous, d decresig for 5x 3 c dx lim dx 5x 3 5 x 3 5 dx lim c c 5 5 x 3 c lim l 5 x 3 c 5 lim l 5 c 3 l 8 5 c Therefore the series diverges
8 The Compriso Test The Compriso Test Let d b be series of positive terms if 2 if b is kow coverget series the series is coverget b is kow diverget series the series is diverget b for ll positive, b for ll positive, I other words, if ll terms of series re less th those of coverget series, the it too must coverge Ad, if ll the terms of series re greter th those of diverget series, the it too diverges Exmple Determie the covergece or divergece of 7 2 Solutio: We kow tht coverges becuse it is geometric series with r 2 Compre the series therefore 7 2 with 2 Ech term of 7 2 is less th 2, 7 2 coverges
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