Convergence of Series $\sum 1/(1+ n^2 x)$ Uniformly (Homework) 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?
 A: For strictly positive $x$ you can already see that each term of the series, except the first one, is bounded by:
$$\frac{1}{1+n^2 x} < \frac{1}{(n-1)^2 x}$$
The series formed by the bounds converges, so by the Weierstrass M-test, you have uniform convergence for $x>0$.
For $x<-1$ you can make the same story:
$$\left|\frac{1}{1+n^2 x}\right| < \frac{1}{(n+1)^2 |x|}$$
The series formed by the bounds converges, so by the Weierstrass M-test, you have uniform convergence for $x<-1$ as well.
The hard part is of course what happens on the open interval $]-1,0[$, you have already excluded the endpoints as well as all the negative inverses of squares of integers within that interval. 
A: HINT:
You have seen a number of discontinuities at certain points.


*

*Can you use the M-test between points of discontinuities? 
1.a If you can then it is good.
1.b If you can not do that - why? Can you do estimates if you remove a part around these points?

*Can you show that the convergence is NOT uniform near these points? 
