This series is pointwise convergent. We have, if is positive that so for all , and the series is normally convergent.
I'm having trouble with this question:
Is the
uniformly convergent over the interval (0,1].
Now I know that it isn't uniformly convergent, and have started my proof with a contradiction.
So assuming for contradiction that it is uniformly convergent, we can say that but I don't know where to go from there. Any help?
Find max f_n(x), by taking the derivative, you'll get x=+/- 1/n^4, x in (0,1] so x=1/n^4
Placing that x=1/n^4 back to f_n(x). You will get f_n(1/n^4)=1/n^2
sum 1/n^2 is converges.
{n^2x}/{1+n^4x^2} < 1/n^2 for all x in (0,1], now the rest comes from Weierstrass test.