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 $C\left(\tfrac 15\right)$?
My mind is blank and I don't know how to solve this problem. Solutions are greatly appreciated!
Note that from the link provided $c_n $ is simply $2^{n-1}$ where $n \in \mathbb{N}$ and from the given problem statement that $c_0 = 1$. Next we can continue to see that our summation will be $$C\Big(\frac{1}{5}\Big) = c_0 + \sum_{n = 1}^{\infty} 2^{n -1}\Big(\frac{1}{5}\Big)^{n} $$ which can be written as $$C\Big(\frac{1}{5}\Big) = 1 + \Big(\frac{1}{5}\Big)\sum_{n = 1}^{\infty} 2^{n - 1}\Big(\frac{1}{5}\Big)^{n-1} = 1 + \Big(\frac{1}{5}\Big)\sum_{n = 1}^{\infty} \Big(\frac{2}{5}\Big)^{n-1}$$ which leads to the solution $$C\Big(\frac{1}{5}\Big) = 1 + \Big(\frac{1}{5} \Big) \Bigg(\frac{1}{1-\frac{2}{5}}\Bigg) = \frac{4}{3} $$