# Power series properties proof

Hey guys,

I've been trying to work out this question,

http://img189.imageshack.us/img189/2954/asdagp.jpg [Broken]

so the identity theorm is just that if the power series = 0 then the coefficient of the series must be zero.

Im having trouble seeing how that negative has any influence over the n in the x^n term, to make the x's in powers of either odd or even.

If you have f(-x) = f(x) then the series would be like

Ʃa (x-xo)^n = Ʃa (-x-xo)^n = Ʃ(-1)^n a (x+xo)^n

So how does that make the powers only even? Is there somthing crusial that im missing?

Thanks

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tiny-tim
Homework Helper
welcome to pf!

hey linda! welcome to pf! forget xo

"even" and "odd" mean about x = 0 does that make it easier?

Thanks!

Yea that helps, so then

Ʃa (x)^n = Ʃa (-x)^n = Ʃ(-1)^n a(x)^n

So is the trick that the only way this can be true is if all the odd powers of n arn't there since the left side will have + a x, + a x^3,.. and the right-a x,- a x^3,... (for an odd ) which can only be true if they are zero an odd = 0?

tiny-tim
Homework Helper
yes but it's a lot easier if you combine it into one series …

∑ { axn - a(-x)n } = 0 Cool,

Thanks a lot!