Rewriting an asymptotic series as a convergent integral I am given the function
$$ 
f(x) = \sum_{n=1}^{\infty} \frac{\Gamma(2n)}{\Gamma(n)}(-x)^n
$$
and I need to show that it can be rewritten as an integral that is convergent for a range of values of x. I was advised to make the following substitution and then “resum” the series
$$ 
f(x) = \sum_{n=1}^{\infty} \int_{0}^{\infty} \text{dt} \frac{t^{2n-1} }{(n-1)!}e^{-t}(-x)^n
$$
I’m unsure how to proceed however. I see that there is potentially a power series buried in here, but the presence of $x$ is confusing. I was also told I can interchange the signs of the integral and the summation but I can’t see where that comes in. How do I rewrite this integral?
Edit: I am also interested in what “resummation” would mean here since I’m having a hard time finding information about that procedure that would seem to be relevant to this situation. I also don’t understand why interchanging the signs of the integral and summation would be allowed.
 A: 
I am given the function
$$ 
f(x) = \sum_{n=1}^{\infty} \frac{\Gamma(2n)}{\Gamma(n)}(-x)^n
$$


and I need to show that it can be rewritten as an integral that is convergent for a range of values of x. I was advised to make the following substitution...

Presumably the substitution you were advised to make was:
$\Gamma(n) \to (n-1)!$ in the denominator and $\Gamma(2n) \to \int dt e^{-t}t^{2n-1}$ in the numerator, though you don't say so explicitly

and then “resum” the series
$$ 
f(x) = \sum_{n=1}^{\infty} \int_{0}^{\infty} \text{dt} \frac{t^{2n-1} }{(n-1)!}e^{-t}(-x)^n
$$


I was also told I can interchange the signs of the integral and the summation but I can’t see where that comes in. How do I rewrite this integral?...

Symbolically, you want to collect all the pieces of your equation for $f(x)$ that depend on $n$, collect them together (e.g., on the far right), and move the summation sign past the integral and as far right as you can. Doing this, you will see that you need to figure out how to do the sum:
$$
S = \sum_{n=1}^{\infty} \frac{(-t^2 x)^n}{(n-1)!}
$$
Hint:
$$
S = (-t^2 x) \sum_{n=0}^\infty \frac{(-t^2 x)^n}{n!}\;,
$$
and you should be able to recognize the latter sum as a well-known series expansion of a well-known function.
