@ir7: Comes from this: For a nonnegative integer n, a composition of n means a partition in which the order of the parts matters. For example, the compositions of 3 are 3, 2+1, 1+2, and 1+1+1.

Consider the generating function $C(x) = \sum_{n=0}^{\infty} c_nx^n$, where $c_n$ is the number of distinct compositions of n (note that $c_0=1$ by convention).

What is the value of $C\left(\tfrac 15\right)$?

@MathyPerson This should work: $$ C(x) = 1 + x+2x^2+2^2x^3 + 2^3x^4+\ldots$$
$$ = 1 + 2^{-1}[2x+(2x)^2+(2x)^3 + (2x)^4+\ldots ] $$
$$ = 1+ x\frac{1}{1-2x} = \frac{1-x}{1-2x}.$$

@ir7 How would I write the power series representation of the power series $1-2x+2x^2-2x^3+2x^4+....$

I have tried this:
$1-2(x-x^2+x^3-x^4+......)$

That means I need to find $x-x^2+x^3-x^4+...$, but I don't know what that is. However, I do know that $1+x+x^2+x^3+x^4+....$ is $\frac{1}{1-x}.$

Would $x-x^2+x^3-x^4+...$ be $\frac{x}{1-x}$?

Question: How would I write the power series representation of the power series $1-2x+2x^2-2x^3+2x^4+....$

My try: I have tried this:
$1-2(x-x^2+x^3-x^4+......)$

That means I need to find $x-x^2+x^3-x^4+...$, but I don't know what that is. However, I do know that $1+x+x^2+x^3+x^4+....$ is $\frac{1}{1-x}.$

Would $x-x^2+x^3-x^4+...$ be $\frac{x}{1-x}$?