= Exponential series divided by a shifted index
{title2=$\sum_{n\geq0}\frac{x^{n+k}}{(n+k)n!}=\int_0^x t^{k-1}e^t\,dt$}
For a positive integer $k$, multiply the <exponential series> by $t^{k-1}$ and integrate term by term from zero to $x$. Its <uniform convergence> on every compact interval justifies the operation and proves the title identity. The resulting <power series> converges absolutely for every real or complex $x$, by the <ratio test>. Repeated <integration by parts> evaluates the right side as an exponential times a <polynomial>, with its constant chosen to make the integral vanish at zero.
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