(2n)/(n^n) does the infinte series converge?

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Homework Help Overview

The discussion revolves around the convergence of the infinite series defined by the term (2n)!/(n^n). Participants are exploring various methods to analyze the series' behavior.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants have attempted the ratio test and the nth root test, expressing uncertainty about their results. Some question the applicability of the vanishing test and the implications of Stirling's approximation.

Discussion Status

There is ongoing exploration of the series' convergence, with multiple participants sharing their findings. Some suggest that the limit approaches infinity, while others question the conditions necessary for convergence. No consensus has been reached yet.

Contextual Notes

Participants note that for convergence, the limit must be zero, which is under scrutiny given the results of their tests.

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Homework Statement



Sorry for the missing summation signs but could anyone help me investigate the convergance of the following infinite sum with n'th term equal to : (2n)!/(n^n)


Homework Equations





The Attempt at a Solution



I have tried ratio test and n'th root test but failed.
Im not even sure if it passes the vanishing test

would appreciate any ideas. Thanks
 
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Use Stirling's approximation.
 
Well, using ratio test, i got that the limit goes to infinity, looks strange, but, i think that it diverges.
 
using sterlings equationm plus nth root test i get a limit which tends to infinity, but i think you can only conclude something about convergance of the sseries if the limit is real...
 
For the series to converge the limit must be zero, right? It's not.
 

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