Pointwise convergence of $\cos$ functions

Question

I've been getting used to pointwise convergence but I have no Idea how to do harder examples.

$$f(n)=\cos(z/n^5)/n!$$

How would I should that this converges pointwise to $f=0$ on $D(0,1)$. I'd really appreciate the help

Answer 1

The sequence converges uniformly to $0$ in $D(0,1)$. Let $M=\sup_{|z|\leqslant 1}\bigl|\cos(z)\bigr|$. Then$$(\forall z\in D(0,1))(\forall n\in\mathbb{N}):\bigl|\cos(z)\bigr|\leqslant M$$and therefore you can use the fact that $\lim_{n\to\infty}\frac M{n!}=0$ to deduce that your sequence converges uniformly on $D(0,1)$, since$$\left|\frac{\cos(\frac zn)}{n!}\right|\leqslant \frac M{n!}.$$