Question: Any one know how to graph a function defined as an infinite series? I need to graph the function $$f(x)=\sum\_{n=1}^{\infty}\bigg(n^{2}\tan^{-1}(x- n^{2}) +n^{2}\tan^{-1}(x+ n^{2}) \bigg) $$
$x\in \mathbb R$.

**EDIT:**
So, as Peter suggested, we can simplify the function using 
 $\tan^{-1} a + \tan^{-1} b = \tan^{-1}((a + b) / (1 - ab))$
to be
$$f(x)=\sum\_{n=1}^{\infty} n^{2} \tan^{-1}\bigg(\frac{2x}{ 1-(x^{2}-n^{4})}\bigg)$$

Answer: 
Here you can see a plot for $n=6,10,20$. It seems that when we sum up to $n$, we obtain a set of $2n$ vertical sigmoids bounded by the lines $y= \pi/2\cdot x$ and $y=-\pi/2\cdot x$. You can open the image to see it larger. 

$\hspace{0.5 cm}$![enter image description here](https://i.stack.imgur.com/RGLVk.jpg)
If you are able to find an explicit formula for the sum of the series, you use that formula to plot the graph of the sum. If not, you choose $N$ large and plot the sum of the first $N$ terms of the series.

In your example I doubt that there is a simpler expression for the sum. $f$ is odd, so that it is enough to consider $x\ge0$. Since $|\tan^{-1}x|\le\max(|x|,\pi/2)$, we see that the series is uniformly convergent. The sum is discontinuous at the points $x=\pm\sqrt{1+n^4}$, $n\in\mathbb{N}$. Below are the graphs of $8$, $16$ and $32$ terms of the sum, drawn with Mathematica 9.

![enter image description here](https://i.stack.imgur.com/ogv0q.png)