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.

Of course finding a formula for the sum of the series is not needed to solve this.
$$\begin{align}
\cos(x^2) &= 1 - \frac{x^4}{2} + O(x^8)\qquad\text{as }x \to 0
\\
\frac{1}{1-\cos(x^2)} &= 2 x^{-4} + O(1)
\\
\sum_{n=4}^\infty n^5 x^n &= 4^5 x^4 + O(x^5)
\\
\frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty n^5 x^n &= 2\cdot 4^5 + O(x)
\end{align}$$
and the limit is $2\cdot 4^5 = 2048$.
