Power Series Representation of e^(sin(x))

[mouser]

Homework Statement

Find the power series representation of the following:

f(x) = e^(sin(x))

Homework Equations

I know this to be true:

e^x = (Inf Sum, n=0) (x^n/n!)

The Attempt at a Solution

So, in substituting sin(x) for x, I get:

e^(sin(x)) = 1 + sin(x)/1 + sin(x)/2 + sin(x)/6 + ...

or

e^(sin(x)) = (Inf Sum, n=0) ((sin(x))^n/n!)

Not sure where I'm going wrong here, so a nudge in the right direction would be greatly appreciated. Thanks in advance!

 

[mouser]

Does anyone have any advice here? I have an exam tomorrow, for which I am almost fully prepared. With my luck though, this problem will be on the exam and I'll be screwed!

Any help would be much appreciated!

 

[mouser]

Also, by (Inf Sum, n=0), I meant $$\sum$$

Thanks

 

[Dick]

I think everybody knows what you mean. You presented the problem pretty clearly. But I don't know how to express the nth term of your series in any nice closed form. If nobody answered they probably don't either. It's pretty easy to find the first few terms by expanding sin(x) as a series. But I don't know the general term.

 

[Defennder]

In short, we have to simplify this:

$$\sum_{r=0}^{\infty}\frac{1}{r!}(\sum_{t=0}^{\infty} \ \frac{(-1)^tx^{2t+1}}{(2t+1)!})^r$$

That looks pretty horrendous to begin with.

 

[BrendanH]

Maybe that IS the answer...

 

[BrendanH]

Except, incorporate the 'r' as t's