Writing $1+3x^2+8x^4+21x^6+\cdots$ as a power series representation

Question

How would I write the power series $$1+3x^2+8x^4+21x^6+\cdots$$ as a power series representation (something neat similar to $\frac{1}{1-x}$)?

This reminds me of the power series $1+x^2+x^4+x^6+\cdots$ where the power series representation for that is $\frac{1}{1-x^2}$, but how would I add the Fibonacci numbers as coefficients into that?

Hints only!!

Answer 1

Let $f(x) = 1+2x+3x^2+5x^3+8x^4+13x^5+21x^6+\cdots$.

You probably already know a closed form for $f(x)$.

Then, $f(-x) = 1-2x+3x^2-5x^3+8x^4-13x^5+21x^6-\cdots$.

Do you see how to get the series you want from $f(x)$ and $f(-x)$?

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To get a closed form for $f(x)$ try combining the following equations in a way that leaves a finite number of terms on the right side:

$f(x) \ \ \ \ = 1+2x+3x^2+5x^3+8x^4+13x^5+21x^6+\cdots$

$xf(x) \ \ = \ \ \ \ \ \ \ 1x+2x^2+3x^3+5x^4+ \ \ 8x^5+13x^6+\cdots$

$x^2f(x) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1x^2+2x^3+3x^4+ \ \ 5x^5+ \ \ 8x^6+\cdots$

Answer 2

Hint: Show that if $A(x) = a_0 + a_1 x + a_2 x^2 + \cdots $, then:

$$\frac{A(x) + A(-x)}{2} = a_0 + a_2 x^2 + \cdots $$

Can you apply this to your series?

Answer 3

Hint 1: What is the generating function for the Fibonacci numbers?

Hint 2: How would you get every other term of a power series? (Big hint: What does $f(x)\pm f(-x)$ do?)