General hints on convergences of series of functions

Question

Study the pointwise and uniform convergence of the following series of functions: a)$$\sum_{n=1}^{\infty}\dfrac{1}{2^n}\cos(3^nx)$$ b)$$\sum_{n=1}^{\infty}\dfrac{1}{n^s}, s\in[1+h,\infty), h>0$$ c)$$\sum_{n=1}^{\infty}\dfrac{x}{n^\alpha(1+xn^2)}$$ Could you please give me some general hints on how to find the convergence of infinite seris of functions? I know Weirstrass theorem and the definition of uniform and pointwise convergence. But I've just learned it today so I am pretty confused. Thank you!

Answer 1

For example:

(1) $\;|\cos(3^nx)|\le 1\;,\;\;\forall\,x\;$

(2) $\;\forall\,s\in[1+h,\,\infty)\;,\;\;\cfrac1{n^s}\le\cfrac1{n^{1+h}}\;$

Answer 2

Hint

for the third.

if $x=0$, it converges.

if $x\neq 0$ it converges if $\alpha>-1$

since

$$\frac{x}{n^\alpha(1+xn^2)}\sim \frac{1}{n^{2+\alpha}}\; ( n\to +\infty)$$