# October 4 Handout.pdf - MATH 21C Discussion Handout...

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MATH 21C Discussion HandoutConvergence Tests for SeriesOctober 4(Updated October 11, 2016)1Useful factsGeometric series:Letabe thefirst termandrbe thecommon ratioof a geometricprogression{ark-1}, k1. The following then hold.nXk=1ark-1=a(rn-1)r-1=a(1-rn)1-rark-1converges if and only if|r|<1 andXk=1ark-1=a1-r.Telescoping series: Suppose{ak}is a sequence and there is a functionfwithak=f(k)-f(k+ 1) for allk1. Then, for all 1m < n, we havenXk=mak=f(m)-f(n+ 1).In addition, iflimn→∞f(n) = 0, thenakactually converges, andXk=1ak=f(1)Analogous laws for integrals carry over to laws for series, eg. addition, subtraction of two series,multiplication by constants, splicing or appending two series.(n-th Term)Divergence Test: Ifanconverges, then{an}converges to 0. Equivalently, if{an}DOES NOTconverge to 0, thenandiverges.WARNING:If the limit of terms is 0, then try a different test!Integral Test: Let{an}be a sequence, wherean=f(n) and there is a positive integerNwherefis continuous, non-negative and decreasing function ofxfor allxN. Then, eitherZNf(x)dxandanboth converges or both diverges.p-Series Test:X1npconverges if and only ifp >1.Adding, deleting or modifying finitely many summands do not affect convergence or divergenceof a series.1
Comparison Test: Suppose{an},{cn}and{dn}are sequences with nonnegative terms.Ifancnfor alln > Nandcnconverges, thenanconverges too.Ifandnfor alln > Nanddndiverges, thenandiverges too.Limit Comparison Test: Suppose{an}and{bn}are sequences with positive terms for allnN. Letlimn→∞anbn=c(a) Ifc >0, thenanandbneither both converge or both diverge.(b) Ifc= 0 andbnconverges, thenanconverges too.(c) Ifcisandbndiverges, thenandiverges too.A seriesanconverges absolutelyif the series|an|converges.Absolute Convergence Test?:If a seriesanconverges absolutely, equivalently|an|converges, then the original seriesanmust converge.Root Test: Letanbe a series and suppose thatlimn→∞np|an|=r(a) Ifr <1, then the series converge absolutely.(b) Ifr= 1, then try a different test!(c) Ifr >1 orris, then the series diverges.Ratio Test: Letanbe a series and suppose thatlimn→∞an+1an=r(a) Ifr <1, then the series converge absolutely.(b) Ifr= 1, then try a different test!(c) Ifr >1 orris, then the series diverges.Alternating Series Test
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