Evaluating $\lim\limits_{x \to 0} \frac1{1-\cos (x^2)}\sum\limits_{n=4}^{\infty} n^5x^n$ I'm trying to solve this limit but I'm not sure how to do it.
$$\lim_{x \to 0} \frac1{1-\cos(x^2)}\sum_{n=4}^{\infty} n^5x^n$$
I thought of finding the function that represents the sum but I had a hard time finding it.
I'd appreciate the help.

If you evaluate the series, you get
$$ -{\frac {{x}^{4} \left( 243\,{x}^{5}-1426\,{x}^{4}+3454\,{x}^{3}-4386
\,{x}^{2}+3019\,x-1024 \right) }{ \left( x-1 \right) ^{6} \left( 1-
\cos \left( {x}^{2} \right)  \right) }}
 $$
$$ \sim \frac{1024x^4}{1-(1-\frac{x^4}{2!}+\dots)}\sim \frac{1024 x^4}{x^4/2!}=2048. $$
