Express this series in terms of the given series A(x)

[ptolema]

Homework Statement

Let A(x) = a₀ + a₁x + a₂x² + a₃x³ + ... = Ʃa_nxⁿ.

Express E(x) = a₀ + a₂x² + a₄x⁴ + ... = Ʃa_2nx²ⁿ.
Do the same for O(x) = a₁x + a₃x³ + a₅x⁵ + ...

Homework Equations

A(x) = a₀ + a₁x + a₂x² + a₃x³ + ... = Ʃa_nxⁿ

The Attempt at a Solution

So I tried using A(x²) as a starting point, but got stuck on how to get the coefficients of A(x²) = a₀ + a₁x² + a₂x⁴ + a₃x⁶ + ... to match those of E(x).
How would I begin expressing E(x) and O(x) in terms of A(x)? Could there be something I could do with sin or cos?

 

[awkward]

Think about A(-x). What is its series?

 

[ptolema]

Think about A(-x). What is its series?

A(-x) = a₀ - a₁x + a₂x² - a₃x³ + ...
I see now. So when I add A(x)+A(-x), I get 2*E(x). Similarly, A(-x)-A(x) = -2*O(x). Thanks, your hint was a huge help!