# Find the expansion of this term

1. Sep 18, 2014

### utkarshakash

1. The problem statement, all variables and given/known data
IF $e^{m \arctan x}=a_0 + a_1x + a_2x^2 + a_3x^3.......+a_nx^n+..........$
prove that $(n+1)a_{n+1} + (n-1)a_{n-1}=ma_n$
and hence obtain the expansion of $e^{m \arctan x}$.

2. Relevant equations

3. The attempt at a solution
$$e^{m \arctan x} = 1+m \arctan x + (m \arctan x)^2/2! + (m \arctan x)^3/3! + ..........$$
But I need to simplify this expansion in terms of x and I'm clueless how to do that.

2. Sep 18, 2014

### Simon Bridge

How did you get the terms of that series?

3. Sep 18, 2014

### Staff: Mentor

The Maclaurin series expansion of emy is
$$1 + my + \frac{(my)^2}{2!} + \dots + \frac{(my)^n}{n!} + \dots$$

Replace y with arctan(x) to get the expansion that utkarshakash shows.

I've been looking at this problem, but no joy as yet.

4. Sep 18, 2014

### Simon Bridge

You cannot just substitute any arbitrary function for y ... consider: what if y=ln|x|/m ?
You have to work out the series coefficients by actually doing the differentiation. It's not as hard as it's, at first, seeming.

5. Sep 18, 2014

### utkarshakash

Got it! Thanks a lot!

6. Sep 18, 2014

### Simon Bridge

No worries - if a shortcut does not get you where you need to be, try going the long way round ;)