Power Series Odd Function Proof

[harrietstowe]

Homework Statement

Suppose that f(x)= summation a_n 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

 

[SammyS]

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?

 

[Dick]

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)?

 

[Gib Z]

For odd functions, $f(-x) + f(x) = 0$. Using that will tell you $a_0 + a_2 x^2 + a_4 x^4 \cdots = 0$.