# Math Help - Power Series Help

1. ## Power Series Help

I have a Question in regards to a Power series questions that I have almost completed.

My Question looks like this:
The function f(x)= 5xarctan(4x) Find:
C1: Which I found to be 0
C2: Which I found to be 0
C3: Which I found to be 20
C4: WHICH I CANT FIGURE OUT, I thought it was 20x^4/3 But it is not right.
As well Find R: Which I cant figure out...

If anyone could help that would be great!

2. $\tan^{-1} \mu = \sum_{n=0}^{\infty} \frac{\mu^{2n+1} (-1)^n}{n}$ for all $-1 < \mu \leq 1$.

This means if $-1 < 4x \leq 1 \implies -\frac14 < x \leq \frac14$ then,
$\tan^{-1} 4x = \sum_{n=0}^{\infty} \frac{4^{2n+1} x^{2n+1} (-1)^n}{n}$.

Thus,
$5x\tan^{-1} 4x = \sum_{n=0}^{\infty} \frac{5\cdot 4^{2n+1} x^{2n+2} (-1)^n}{n}$.