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Is there a closed form solution to the series $\sum ^{\infty }_{n=0}\dfrac {1}{n^{n}}$ ? I ran a python program and tried to use ries to find a closed form with no luck. Thanks

Husnain Raza
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1 Answers1

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I suppose you mean the upper limit of the summation is an integer rather than infinity. In that case, I think there is no general formula.

Reza Lotfalian
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  • Why would you assume the question is not as stated? Do you have an answer if the upper limit is $\infty$? – robjohn Oct 23 '16 at 02:24
  • Because the question provides values for lower and upper bounds. In that case, the answer is a number, not a closed form formula. It's like when you specify the limits of an integral and you get a number, compared to when at least one of limits of the integral is a parameter. In the latter, you get a parametric solution.

    The value of the summation as stated in the question is 1.291... as given in the link carmichael561 shared above.

     – Reza Lotfalian Oct 23 '16 at 02:32
  • One can ask for a closed form of $\sum\limits_{k=1}^\infty\frac1{k^2}$. The answer would be $\frac{\pi^2}6$. No parameters. In any case, where are the upper and lower bounds given? – robjohn Oct 23 '16 at 02:44
  • Thanks for the point regarding the closed form solutions.

    I have been careless in using "bound" instead of "limit". In my native language, the two words mean the same. That's why I've been sloppy. I'll make sure there would be no such mistakes in my future comments.

     – Reza Lotfalian Oct 23 '16 at 03:50