I'm trying to prove that the series below converges to 1 and I noticed it looked strikingly similar to a probability distribution I once saw. My question is twofold:

  1. Can anyone identify the distribution? I can't seem to, for the life of me, remember. I'm 90% sure this is a probability distribution (or some form of one) but I may be wrong.
  2. Does anyone have any hints for proving this convergence? I don't necessarily want an answer, just some pointers. I don't think this requires any complex mathematics beyond an early graduate course in analysis.

$$ \sum_{j=1}^{\infty}{\frac{e^{-j}j^{j-1}}{j!}}= 1 $$

Thank you!

  • 2
    $\begingroup$ The identity follows by setting $x=1$ in the well-known formula $$\sum_{j=1}^\infty \frac{(xe^{-x})^j j^{j-1}}{j!} = x,$$ which is equivalent to the Taylor series for the Lambert W function that Robert Israel mentioned. A simple proof is by expanding the left side in powers of $x$ and using the fact that the $n$th difference of a polynomial of degree less than $n$ is 0. Another simple proof was given by Noam Elkies, math.harvard.edu/~elkies/Misc/abel.pdf. $\endgroup$
    – Ira Gessel
    Commented Dec 25, 2016 at 17:31
  • $\begingroup$ @IraGessel That's a really nice proof. Thank you. If you'll make your comment into an answer I will gladly up vote it. $\endgroup$
    – gowrath
    Commented Dec 25, 2016 at 22:03

1 Answer 1


1) Presumably you're talking about the Poisson distribution, but I don't think that's useful.

2) Possibly more useful is that this is related to the Taylor series for the Lambert W function.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.