Prove that diverges for . Also find the closed-form of the series for x in the radius of convergence.
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As for the closed form can be obtained from applying the binomial series to .
Originally Posted by mathwizard Prove that diverges for . Also find the closed-form of the series for x in the radius of convergence. Therefore when The series converges
Originally Posted by Mathstud28 Therefore when The series converges Huh? What test are you using to determine the convergence (are you using Stirling's approximation for ?)? The ratio test seems to yield an inconclusive result (i.e., the limit is 1).
For I think you can use Abel's theorem. It would mean the series is continous at which is a problem because it is equal to when and therefore not defined at .
Originally Posted by mathwizard Huh? What test are you using to determine the convergence (are you using Stirling's approximation for ?)? The ratio test seems to yield an inconclusive result (i.e., the limit is 1). Correct me if I am wrong but, Let Well and Thus the series converges by the alternating series test
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