What are some of the more advanced convergence / divergence verification tests for infinite series'?
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From: Cogito on 10 May 2010 19:37 I have the commonly taught ones I consider trivial... all colleges teach these and all calculus texts do as well. Trivial. "limit of the sequence", "nth term test", or "divergence test" geometric p-series telescoping series ratio test absolute convergence tests alternating series test roots test cauchy integral test direct comparison limit comparison Then I also know these tests that I had to find on my own. In my own research I discovered... ratio limits extremum test (in case ratio test L=1) roots limits extremum test (in case roots test L=1) Raabes test (in case ratio test inconclusive) Bertrands test Gauss's test Abel's test Direchlets test Kummer's test Ermakoff's test Cauchy Condensation test These I know. Im looking for others, if there are any. No need to explain them, just give me something to look up. You cant look up a something if you dont know what to type into the search bar. Unfortunately, all of my "convergence tests" Google searches turn up only the trivial cases from the first list, and sometimes if I look hard enough I might find the non-trivial ones from the second list too... but I cannot find any others (except for Fourier series tests - but I dont care for Fourier series at this point)
From: Achava Nakhash, the Loving Snake on 12 May 2010 08:33
On May 10, 4:37 pm, Cogito <cogitoergocogito...(a)gmail.com> wrote: > I have the commonly taught ones I consider trivial... all colleges > teach these and all calculus texts do as well. Trivial. > "limit of the sequence", "nth term test", or "divergence test" > geometric > p-series > telescoping series > ratio test > absolute convergence tests > alternating series test > roots test > cauchy integral test > direct comparison > limit comparison > > Then I also know these tests that I had to find on my own. In my own > research I discovered... > ratio limits extremum test (in case ratio test L=1) > roots limits extremum test (in case roots test L=1) > Raabes test (in case ratio test inconclusive) > Bertrands test > Gauss's test > Abel's test > Direchlets test > Kummer's test > Ermakoff's test > Cauchy Condensation test > > These I know. Im looking for others, if there are any. No need to > explain them, just give me something to look up. You cant look up a > something if you dont know what to type into the search bar. > Unfortunately, all of my "convergence tests" Google searches turn up > only the trivial cases from the first list, and sometimes if I look > hard enough I might find the non-trivial ones from the second list > too... but I cannot find any others (except for Fourier series tests - > but I dont care for Fourier series at this point) Wow! Arent' you wonderful! Achava |