Given $\sum_{n \geq 0} a_n x^n$, where $a_n$ is the number of strings of length n all of whose entries equals 1, find a closed form.
If I am correct so far, I have (0, 1, 2, 3, ...) as the counting sequence, equivalent to $(x + 2x^2 + 3x^3 + \cdots)$.
How does one go from sequence to closed form generating function?
Since there is a unique string of length $n$ consisting only of 1s, you have for each $n$ that $a_n = 1$, and so your generating function is $$ \sum_{n=0}^\infty a_nx^n = 1 + x + x^2 + x^3 + x^4 + \cdots $$ This is a geometric series, and it follows that $$ \sum_{n=0}^\infty a_nx^n = \frac{1}{1 - x} $$ is the closed form that you're looking for.