Verify Infinite Series: Proving Cosine Sum and Integrals

[jj1986]

Homework Statement

Show that for all integers n $$\geq$$ 1,
cos(2x) + cos(4x) + ... + cos(2nx) = $$\frac{1}{2}$$ ($$\frac{sin((2n+1)x)}{sin(x)}$$-1)

Use this to verify that
$$\sum_{n=1}^{\infty}(\int_{0}^{\pi}$$ x($$\pi$$-x)cos(2nx)dx) =

$$\frac{-1}{2}\int_{0}^{\pi}$$ x($$\pi$$-x)dx)

Homework Equations

The Attempt at a Solution

I proved the first part of this problem using induction, however I don't see how I can use that to verify the second part. Maybe I can bring the summation into the integral and get the sum of cos(2nx), but I still don't see how that would give me what I need to prove. Any suggestions?

 

[CompuChip]

If you interchange the sum with the integral, you can substitute the identity you have proved.
Then the second part (with the - 1) will give you the answer you want, you will just need to prove that
$$\lim_{n \to \infty} \frac12 \int_0^\pi x (\pi - x) \frac{\sin((2n+1)x)}{\sin(x)}\, dx = 0$$