Given a series of functions, how do we find the interval/region of uniform convergence?? Is there any technique to go about solving this problem??
As for example, consider a series $\sum \frac{\log {n}}{n^{x}}$. The question asks to find the intervals/regions of uniform convergence.
So far, what I have done is prove that the series converges pointwise for $x \gt 1$. But, how do I go ahead and find the intervals/regions of uniform convergence for this particular problem?? This throws me.
M-Test tells you that $\sum f_n(x)$ converges uniformly on a set $A$ if $|f_n(x)| \leq a_n$ for all $x \in A$ and all $n \geq 1$ with $\sum a_n <\infty$.
In this case we can take $a_n=\frac {\ln n } {n^{1+\epsilon}}$ to see that the series converges uniformly on $[1+\epsilon, \infty)$ for any $\epsilon >0$.
The convergence is not uniform on $(1,\infty)$. Let us prove this by contradiction. If the convergence is uniform there would exist $N$ such that $| \sum\limits_{n=n_1}^{n_2} \frac {\ln n} {n^{x}}| <1$ for all $x >1$ whenever $n_2,n_1>N$. If we let $x \to 1$ in this we get a contradiction since the series $\sum \frac {\ln n} {n}$ is divergent.