Problem - 4313
Let $f(x)$ be the generating function for $a_0$, $a_1$, $a_2$, $\cdots$. Find the generating function for $$a_0, a_0 + a_1, a_0+a_1+a_2, \cdots$$
The answer is $$f(x)\sum_{n=0}^{\infty}x^n$$
The given condition implies $$f(x) = \sum_{n=0}^{\infty}a_nx^n$$
Then $$\begin{align*} f(x)\sum_{n=0}^{\infty}x^n=\ &\left(\sum_{n=0}^{\infty}a_nx^n\right)\left(\sum_{n=0}^{\infty}x^n\right) = \sum_{n=0}^{\infty}\left(\sum_{k=0}^{n}a_n\right)x^n\\ \\=\ &\sum_{n=0}^{\infty}\left(a_0 + a_1 +\cdots + a_n\right)x^n\end{align*}$$