How to tell whether the series converges or not?

Question

I have a series given by: $$x\bigg\{\frac{e^{-t}}{2}+\frac{1}{2!} \frac{e^{-t}(t^2-x^2)}{8}+\frac{1}{2!3!} \frac{e^{-t}(t^2-x^2)^2}{32}+\frac{1}{3!4!} \frac{e^{-t}(t^2-x^2)^3}{128}+\ldots\bigg\}$$ I want to know whether this series converges. I mean to say that the term $$\frac{(t^2-x^2)^n}{(n!)^2(2)^{2n+1}}e^{-t}$$ has finite value for $n\rightarrow\infty$, for an arbitrary value of $t.$

Answer 1

After having removed the extra factor $e^{-t}$ and letting $a=t^2-x^2$, we have to determine whether the series $\sum_n b_n$ is convergent with
$$b_n:=\frac{a^n}{2^{2n+1}(n!)^2}.$$ A possibility is the ratio test: the limit of the sequence $\left(b_{n+1}/b_n\right)_{n\geqslant 1}$ is not hard to compute.