# Does #a_n=n^x/(n!) # converge for any x?

The sequence

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For every

Exponential grows faster than power.

Factorial grows faster than exponential.

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The sequence ( a_n = \frac{n^x}{n!} ) converges if and only if the limit of ( a_n ) as ( n ) approaches infinity exists and is finite. This convergence is determined by the value of ( x ).

For any fixed value of ( x ), the sequence ( a_n ) converges if ( x ) is less than or equal to zero. When ( x ) is a non-negative integer or zero, the sequence converges to ( 0 ) as ( n ) approaches infinity.

However, if ( x ) is a positive non-integer, the sequence ( a_n ) diverges as ( n ) approaches infinity. This is because the factorial ( n! ) grows faster than any power of ( n ), causing the terms ( \frac{n^x}{n!} ) to approach zero too slowly for convergence.

Therefore, the sequence ( a_n = \frac{n^x}{n!} ) converges only for ( x \leq 0 ) and diverges for ( x > 0 ).

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