Power Series Odd Function Proof

  • #1

Homework Statement


Suppose that f(x)= summation an x^n for n = 0 to infinity for all x. If f is an odd function, show that a0 = a2 = a4 = ... = 0.


Homework Equations





The Attempt at a Solution


I said to consider sin(x), an odd function. When you do a series expansion only odd terms exist in the series so all even terms are equal to zero. This is because the nth derivative of sin(x) where n is an even number >= 2 always produces some form of cos(x) and when you plug 0 into cos(x) you get 0 and so even terms disappear. I was told that I restated the question and that I need to think harder about odd functions. I don't know how else to answer this. Thanks
 

Answers and Replies

  • #2
SammyS
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That's one EXAMPLE. It doesn't prove it's true for all such functions.

Suppose you have a function for which a0≠0, or a2≠0, or a4≠0, ...

Show that such a function is NOT odd.

What's true if a function is odd?

What's true if a function is NOT odd?
 
  • #3
Dick
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If f(x) is odd then f(0)=0, right? Now do you know that if f(x) is odd then f'(x) is even? And vice versa? Now what about f''(x)? What about the nth derivative of f(x)?
 
  • #4
Gib Z
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For odd functions, [itex] f(-x) + f(x) = 0 [/itex]. Using that will tell you [itex] a_0 + a_2 x^2 + a_4 x^4 \cdots = 0 [/itex].
 

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