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)
Here is a graph of the series by maple 12. To see that there are jumps at some point (look where the vertical segments are), just notice that, the value of the series at $x=4.12$ is positive $9.2885852$ and at the point $x=4.13$ is negative $-3.2263024$.

![enter image description here](https://i.stack.imgur.com/0GzJv.jpg)

This is the plot of $f'(x)$

![enter image description here](https://i.stack.imgur.com/bDRK1.jpg)