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# Limit of (n!/n^n)^(1/n) watch

1. Greetings!

I'm trying to work out the radius of convergence of the function

using the Cauchy root test:

To go about solving this, I set and took the natural logarithm of :

I then used Stirling's approximation, (my mathematician friend laughed at me for doing this, but I'm a physicist and thus love a good approximation ), and so

which simplifies to

and so taking the limit gives . This means the answer to the original limit is given by

and so the radius of convergence is (or at least I think it is!).

The book solution simply states that "Since the th root of tends to as [poorly worded!], the limit of this ratio is that of , namely unity. Thus and the series converges inside the unit circle." I don't really understand where this result comes from, and why it is different to mine! Was my friend right for laughing at my use of Stirling's approximation?

Any help and insights will be much appreciated Thanks!
2. It looks like it'd be easier with the ratio test.

EDIT: I get the same answer as you, using the ratio test (much more easily/rigorously). Wolfram agrees with me (well, agrees that R isn't 1), since using z = 2 the series converges.
3. Thanks for the response! Did you use Stirling when you did the ratio test? I'm wondering if there is a different method (which I'm sure there is) of dealing with the ...
Thanks for the response! Did you use Stirling when you did the ratio test? I'm wondering if there is a different method (which I'm sure there is) of dealing with the ...
I really would avoid using things like Stirling if possible. The n! cancels nicely in the ratio test

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Updated: March 29, 2011
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