Power series representation of the sum of $\sum_{k=0}^\infty(k^3+1) x^k/k!$ So, recently I was searching for, what seemed like a fairly simple one, what function does this sum represent:
$\displaystyle f(x)=\sum_{k=0}^\infty\left(k^3+1\right)\frac{x^k}{k!}$
So the natural thing would be to separate both equations:
$\displaystyle f(x)=\sum_{k=0}^\infty k^3\frac{x^k}{k!}+e^x$
So, with further identification $f(x)$ seemed like a simple answer:
$f(x)=T_3(x)e^x+e^x$
where $T_n$ is the $\,n^{\mathrm{th}}$ Touchard polynomial.
$f(x)=\left(x+3x^2+x^3+1\right)e^x$
What bugged me is the fact that neither I do I understand Touchard Polynomials nor does the question fit the level of knowing them, this is intended as a Second Semester undergrad Calculus II question, so my question is simple, is there a method of finding the sum without having to deal with the Touchard polynomials, is there a way of arriving to the same (or different, maybe I’m wrong) answer without dealing with them?

$$
\begin{align}
\sum_{k=0}^\infty k^3 \frac{x^k}{k!} &= x\sum_{k=1}^\infty k^3 \frac{x^{k-1}}{k!} \\
&=x \sum_{k=0}^\infty (k+1)^2 \frac{x^k}{k!} \\
&=x\left(\sum_{k=0}^\infty (k+1)\frac{x^{k+1}}{k!}  \right)^\prime \\
&=x\left(\sum_{k=1}^\infty k \frac{x^{k+1}}{k!} +xe^x\right)^\prime \\
&=x\left(x^2e^x + xe^x\right)^\prime \\
&=x(x^2 + 3x +1)e^x
\end{align}
$$
