All-Tests-Series - All Tests to Determine an Innite Series is Convergent or Divergent Math 201 Fall 2013 Kamal Aziziheris 1(n-th Term Test If limn an =

All-Tests-Series - All Tests to Determine an Innite Series...

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All Tests to Determine an Infinite Series is Convergent or Divergent Math 201 - Fall 2013 Kamal Aziziheris September 16, 2013 1. (n-th Term Test) If limn→∞an6= 0, thenn=1anisDivergent.2. (Geometric Series) Supposeaandrare fixed numbers. If|r|<1,thenXn=1arnisConvergenttoa1-r.Otherwise, it isDivergent.3. (Telescopic Series) Assume that for alln,an=bn-bn+1, where{an}and{bn}are two sequences. ThenXn=1an=b1-limn→∞bn.4. (Integral Test) Suppose that for alln,an=f(n), wherefis acontinuous, positive, decreasing function ofxfor allxN(Nis apositive integer). Thenn=NanandRNf(x)dxboth convergeorboth diverge.5. (Comparison Test) Letanandbnbe series with nonnegativeterms. Suppose that for some integerN,anbnfor allnN. Then:Ifbnconverges, thenanconverges.Ifandiverges, thenbndiverges. 1
6. (Limit Comparison Test) Suppose thatan, bn>0 for allnN,whereNis a fixed positive integer. IfL= limn→∞anbn,thenIf 06=L6=(call it the“Ideal Case”), thenanandbnboth convergeorboth diverge.IfL= 0 andbnconverges, thenanconverges.IfL=andbndiverges, thenandiverges.7. (n-th Root Test) Suppose thatan>0 for allnN, whereNis afixed positive integer. IfL= limn→∞nan,thenIfL <1, thenanconverges.IfL >1, thenandiverges.IfL= 1, then the test fails.8. (Ratio Test) Suppose thatan>0 for allnN, whereNis a fixedpositive integer. IfL= limn→∞an+1an,thenIfL <1, thenanconverges.IfL >1, thenandiverges.IfL= 1, then the test fails. 2

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