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

##### 2 Answers

In fact you just need to remember that

So the answer is yes

but here the demonstration

study

its

the numerator don't depend of n so we study the denominator

this limit is pretty hard, by intuition you can imagine it will be

you can rewrite

so we need to study

hopefully we have this awesome approximation

so we study

which is

so we proved that

so

So by Cauchy-Hadamard Theorem

radius of convergence

Also, the Maclaren power series representation for the series

Alternatively, one could show that this is a Cauchy sequence since

Hence it converges since by a theorem, any sequence