Sequences and Series of Functions

1. Apr 12, 2010

Fiz2007

1. The problem statement, all variables and given/known data

Let sum of a sub k
be an absolutely convergent series.

a. Let f be the function defined by f(x) = sum of (a sub k) * sin(kx). Prove that:

the integral from 0 to pi/2 of f = sum of (a2k-1 + a4k-2)/(2k-1)
2. Relevant equations

I already showed that f(x) converges uniformly using the Weirstrass M theorem

3. The attempt at a solution

I'm completely stuck. I understand convergence of sequences and proving uniform continuity of sequences, but when we begin to use the summation with the sequence of functions, I am totally lost. Any help would be greatly appreciated.

2. Apr 12, 2010

lanedance

have you tried evaulating the integral explicitly?

3. Apr 12, 2010

lanedance

also to write in tex see below
$$\int^{\pi/2}_0 dx f = \int^{\pi/2}_0 dx \sum a_k sin(kx)$$