Is (n!/n^n)*exp^n a Convergent or Divergent Summation?

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The discussion centers on the convergence of the summation from n=0 to infinity for the expression (n!/n^n)*exp^n. Participants assert that applying Stirling's formula reveals that the individual terms of the series grow larger, indicating divergence. Specifically, the approximation of n! using Stirling's formula leads to terms that do not approach zero, confirming that the series diverges. Thus, the conclusion is that the summation is divergent.

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1)summation for n=0 to infinity for (n!/n^n)*exp^n. Can anyone help to prove whether this is convergent or divergent?
 
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adiputra said:
1)summation for n=0 to infinity for (n!/n^n)*exp^n. Can anyone help to prove whether this is convergent or divergent?

Can you refer to Stirling's formula? When you use it, the n term in the sequence is approximately (2(pi)n)1/2. The series could never converge, with the individual terms getting larger.
 
thank a lot
 

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