Guys I got across this question in my last chapter of real analysis book. Prove that $\sum $$(x^2+n)\over(x^2+n^3)$ is not uniformly convergent on the domain $\Bbb R$.
This is Book's Answer:
When $x=n^2$, there is a term in the series, after the nth, which is greater than $1\over 2$. So altough the series is convergent for each value of x by comparison with $\sum 1/n^2$, the greatest difference between the nth partial sum (i. e.$f_n$) and limit function is not null.
The first statement in answer that book made is where I don't understand. Why is $x=n^2$ even choosed? How it came? And what does it mean by after nth term of it is greater than $1\over 2$?
For uniform convergece of the series $\sum f_n(x)$ a necessary condition is $f_n(x) \to 0$ uniformly. This means $\sup_x |f_n(x)| $ must tend to $0$. This forces $f_n(x_n)$ also to tend to $0$ for any sequence of numbers $(x_n)$ in the domain. So for proving that a series is not uniformly convergent you try to find $x_n$'s so that $f_n(x_n)$ does not tend to $0$.
The choice of $x_n$ used in the proof depends on the particular series. There is no general method. When $f_n$ is a ratio of two simple functions as in this case you try to choose $x_n$ so that the numerator and the denominator tend to $0$ or $\infty$ at the same rate so that the ratio tends to some non-zero number. Basically you have to start guessing what $x_n$'s might work.