The task is to specify convergence of infinite function series (pointwise, almost uniform and uniform convergence):
a) $\displaystyle \sum_{n=1}^{+\infty}\frac{\sin(n^2x)}{n^2+x^2}, \ x\in\mathbb{R}$
b) $\displaystyle\sum_{n=1}^{+\infty}\frac{(-1)^n}{n+x}, \ x\in[0;+\infty)$
c) $\displaystyle\sum_{n=0}^{+\infty}x^2e^{-nx}, \ x\in(0;+\infty)$
I know basic facts about these types of convergence and Weierstrass M-test, but still have problems with using this in practise..
Best Answer
First observe that each of the series converges pointwise on its given interval (using standard comparison tests and results on $p$-series, geometric series, and alternating series.
Towards determining uniform convergence, let's first recall the Weierstrass $M$-test:
It is worthwhile to consider the heart of the proof of this theorem:
Under the given hypotheses, if $m>n$, then for any $x\in I$ $$\tag{1} \bigl| f_{n+1}(x)+\cdots+f_m(x)\bigr| \le| f_{n+1}(x)|+\cdots+|f_m(x)\bigr| \le M_{n+1}+\cdots M_n. $$ So if $\sum M_n$ converges, we can make the right hand side of $(1)$ as small as we wish. Noting that the right hand side of $(1)$ is independent of $x$, we can conclude that $\sum f_n$ is uniformly Cauchy on $I$, and thus uniformly convergent on $I$.
Now on to your problem:
To apply the $M$-test, you have to find appropriate $M_n$ for the series under consideration. Keep in mind that the $M_n$ have to be positive, summable, and bound the $|f_n|$. Sometimes they are easy to find, as in the series in a). Here note that for any $n\ge 1$ and $x\in\Bbb R$, $$ \biggl| {\sin(n^2x)\over n^2+x^2}\biggr|\le {1\over n^2}. $$ So, take $M_n={1\over n^2}$ and apply the $M$-test. The series in a) converges uniformly on $\Bbb R$.
Sometimes finding the $M_n$ is not so easy. This is the case in c). Crude approximations for $f_n(x)=x^2e^{-nx}$ will not help. However, we could try to find the maximum value of $f_n$ over $(0,\infty)$ and perhaps this will give us what we want. And indeed, doing this (using methods from differential calculus), we discover that the maximum value of $f_n(x)=x^2e^{-nx}$ over $(0,\infty)$ is ${4e^{-2}\over n^2}$. And now the road towards using the $M$-test is paved...
Sometimes the $M$-test doesn't apply. This is the case for the series in b), the required $M_n$ can't be found (at least, I can't find them). However, here, the proof of the $M$-test gives us an idea. Since the series in b) is alternating (that is, for each $x\in[0,\infty)$, the series $\sum\limits_{n=1}^\infty{(-1)^n\over x+n}$ is alternating), perhaps we can show it is uniformly Cauchy on $[0,\infty)$.
Indeed we can:
For any $m\ge n$ and $x\ge0$ $$\tag{2} \Biggl|\,{(-1)^n\over n+x}+{(-1)^{n+1}\over (n+1)+x}+\cdots+{ (-1)^m\over m+x}\,\Biggl|\ \le\ {1\over n+x}\le {1\over n}. $$ The term on the right hand side of $(2)$ is independent of $x$ and can be made as small as desired. So, the series in b) is uniformly Cauchy on $[0,\infty)$, and thus uniformly convergent on $[0,\infty)$.