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.

Note that the first term is $$\frac 1 {1-(1-x^4/2+O(x^8))} = \frac 2 {x^4}(1+O(x^4))$$ and that the second term is $$4^5 x^4(1+O(x))$$
(The question keeps changing; this is for the $$\cos x^2$$ denominator.)
