Find the expansion of this term

[utkarshakash]

Homework Statement

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}.

Homework Equations

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.

 

[Simon Bridge]

How did you get the terms of that series?

 

[Mark44]

How did you get the terms of that series?

The Maclaurin series expansion of e^my 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.

 

[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.

 

[utkarshakash]

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.

Got it! Thanks a lot!

 

[Simon Bridge]

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