1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Power series properties proof

  1. Mar 22, 2012 #1
    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?

    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Mar 22, 2012 #2


    User Avatar
    Science Advisor
    Homework Helper

    welcome to pf!

    hey linda! welcome to pf! :smile:

    forget xo

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

    does that make it easier?
  4. Mar 22, 2012 #3

    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?
  5. Mar 22, 2012 #4


    User Avatar
    Science Advisor
    Homework Helper

    yes :smile:

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

    ∑ { axn - a(-x)n } = 0 :wink:
  6. Mar 22, 2012 #5

    Thanks a lot!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook