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Power series properties proof

  • Thread starter linda300
  • Start date
  • #1
61
3
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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Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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welcome to pf!

hey linda! welcome to pf! :smile:

forget xo

"even" and "odd" mean about x = 0 :wink:

does that make it easier?
 
  • #3
61
3
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?
 
  • #4
tiny-tim
Science Advisor
Homework Helper
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yes :smile:

but it's a lot easier if you combine it into one series …

∑ { axn - a(-x)n } = 0 :wink:
 
  • #5
61
3
Cool,

Thanks a lot!
 

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