Limit of power series, 2 variables In an assignment, I've run into the following problem.
$$\lim_{x\to\infty} \sum ^\infty_{n=3} \frac{n\cdot x^{n-2}\cdot(-1)^n\cdot(n-1)}{(2n)!}$$
I really hope someone can steer me the right way, thanks!

We have that
$$\cos x=\sum ^\infty_{n=0} \frac{(-1)^n\cdot x^{2n}}{(2n)!}\implies \cos \sqrt x=\sum ^\infty_{n=0} \frac{(-1)^n\cdot x^{n}}{(2n)!}$$
$$-\frac{\sin \sqrt x}{2\sqrt x}=\sum ^\infty_{n=1} \frac{(-1)^n\cdot n x^{n-1}}{(2n)!}$$
$$\frac{\sin \sqrt x-\sqrt x \cos \sqrt x}{4\sqrt {x^3}}=\sum ^\infty_{n=2} \frac{(-1)^n\cdot n (n-1)x^{n-2}}{(2n)!}$$
therefore
$$\sum ^\infty_{n=3} \frac{(-1)^n\cdot n (n-1)x^{n-2}}{(2n)!}=\frac{\sin \sqrt x-\sqrt x \cos \sqrt x}{4\sqrt {x^3}}-\frac1{12}$$
