This is from the book Principles of Mathematical Analysis by Rudin, number 4 of chapter 7. It says consider
$$ f(x) = \sum\limits_{n=1}^{\infty}{ 1/(1+ n^2 x) } $$
The question asks:
(1) For what values of x does the series converge absolutely. We got that the series converges when x $\not=$ 0 & x $\not= -1/k^2 $ when k is an integer since, there is a discontinuity when n reaches the value $k^2$ .
However we don't understand how to do any of the following questions asked. Any hints would be greatly appreciated. We were told that this problem was suppose to be fairly hard for its position in the problem set (ie. 4th question in the Rudin book).
(2) What interval does it converge uniformly?
(3) On what intervals does it fail to converge uniformly ?
(4) Is f continuous wherever the series converges?
(5) Is f bounded?