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