I have a series of functions given by sin[x/(n^2)]. I know that it is convergent, but not uniformly convergent, but I'm not sure how to show these properties.
For convergence, I see that abs(sin[x/(n^2)]) <= abs(x/(n^2)) = abs(x)/n^2.
I believe that I can say the series abs(x)/n^2 converges since its limit is zero for all x in R. And by the Comparison Test, the original series of functions converges. I don't think this is correct, so how would I show it converges?
Also, I know it doesn't converge uniformly (from the back of the book), but we don't seem to have any tests to prove this (just to show that it is uniformly convergent). How would I show this (that it is not uniformly convergent)?
Thanks for your help. It's greatly appreciated.
I omitted the summation symbol and notation just because I don't know how to write it on the computer.