Convergence of Sum 1/n(n+1) * (sin(x))^n

[stukbv]

Homework Statement

Sum 1/n(n+1) * (sin(x))^n .
Show this converges for all x in the reals.
Find with proof an interval on which it determines a differentiable function of x together with an expression of its derivative in terms of standard functions.

Homework Equations

The Attempt at a Solution

I have showen convergence via a comparison to sum 1/n(n+1) so this is okay,
I think now we let x=sinx and write it like ;
Sum 1/n(n+1) * x^n .
Differentiating this gives you 1/(n+1) * x^n-1 now basically am stuck again :(

Thankyou!

 

[micromass]

Hi stukbv!

I have showen convergence via a comparison to sum 1/n(n+1) so this is okay,

Sounds good!

I think now we let x=sinx and write it like ;
Sum 1/n(n+1) * x^n .
Differentiating this gives you 1/(n+1) * x^n-1 now basically am stuck again :(

No, now you're calculating a different derivative. You really need to calculate the derivative of

\sum_{n=0}^{+\infty}{\frac{1}{n(n+1)}\sin^n(x)}