Is there a simple expression or approximation for the following series of exponential integrals:

$\sum_{k=1}^\infty E_1(k \nu)$

I believe this can be approximated by $\int_\nu^\infty E_1(x)/\nu \, dx$ for small enough $k$, but I'm interested in a closed form, or at least an estimate for the error from the approximation above.

  • 1
    $\begingroup$ It is $\int_0^\infty \lfloor x\rfloor \frac{e^{-x \nu}}{x}dx$. Expanding $\lfloor x\rfloor - x$ in a Fourier series you'll obtain a closed form in term of the Hurwitz zeta function. $\endgroup$ – reuns Oct 14 '17 at 6:07
  • $\begingroup$ Make this an answer. It won't cut you off if you hit return. $\endgroup$ – marty cohen Oct 14 '17 at 6:10

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