Sequences and Series of Functions

[Fiz2007]

Homework Statement

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 (a_2k-1 + a_4k-2)/(2k-1)

Homework Equations

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

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.

 

[lanedance]

have you tried evaulating the integral explicitly?

 

[lanedance]

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