Solving Basis Functions Homework w/ Constants A_n & B_n

[Fizz_Geek]

Homework Statement

Given x in the interval [0, $$\pi$$], let $$\phi$$$$_{0}$$(x) = 1, and $$\Phi$$$$_{n}$$ (x) = sin ((2n-1)x).

Show that there are constants:
{A$$_{n}$$}$$^{n=0}_{\infty}$$ and {B$$_{n}$$}$$^{n=0}_{\infty}$$

such that:

$$\sum$$$$^{n=0}_{\infty}$$A$$_{n}$$$$\phi$$$$_{n}$$=$$\sum$$$$^{n=0}_{\infty}$$B$$_{n}$$$$\phi$$$$_{n}$$

But A$$_{n}$$ $$\neq$$ B$$_{n}$$ $$\forall$$n


All the n's should be subscripts. None are powers.

Relevant equations

I really don't know where to start. Any push in the right direction would be greatly appreciated.

 

[JonF]

note $$\Phi$$_n won't give you cyclic results unless your x is a rational multiple of pi, making this a bit tricky.

Luckily -1<=$$\Phi$$_n <=1.

So I would define try something like this.

Define A_n = a_n , where a_n *$$\Phi$$_n = (-1)^n/n^2
This would mean the A_n series converges.

Then i would define B_n so it reorders the terms of your A_n series. So it would converge and be equal.

 

[Fizz_Geek]

Thanks for the reply!

I thought to do something like that, and I can understand that both series would converge, but how would they be equal?

 

[JonF]

They would only be equal as you took the limit to infinity, which is all your proof requires. Since the A_n series is absolutely convergent all of it's rearrangements will be equal. Note, you need to make A_n absolutely convergent, or this won't be true.