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} \foralln


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.