Exponential series divided by a shifted index (${\sum }_{n\ge 0}\frac{x^{n+k}}{(n+k)n!}={\int }_0^x{t}^{k-1}{e}^{t}dt$)

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.