Question

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

Answer 1

The sequence `a_n->0` for any fixed `x`

Explanation:

For any fixed `x = x_0` there is a `n_0` such that for `n>n_0` the sequence `a_n->0`. The converse is easy to obtain. Given a `n_0` then fixing a `x_0 = log_e(n_0!)/log_en_0`, for `n>n_0` then `a_n(x_0)->0`

Answer 2

For every `x`, `lim_(nrarroo)n^x/(n!) = 0`

Explanation:

Exponential grows faster than power.

`AAx` `lim_(nrarroo)n^x/(e^n) = 0`

Factorial grows faster than exponential.

`lim_(nrarroo)e^n/(n!) = 0`

The product must also go to `0`