I would like to ask if anyone would help me to find a general formula for the sum of m members of the following series. \begin{equation} \sum_{n=1}^{m} n^n = 1^1+2^2+\cdots + m^m \end{equation}
Thank you in advance.
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4$\begingroup$ AFAIK, there is no known general formula for this sum. $\endgroup$– Ayman HouriehCommented Aug 25, 2019 at 10:52
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7$\begingroup$ Nice little rule of thumb: If OEIS has your sequence, but doesn't have a closed expression for it, chances are there are no closed form expressions. Of course, absence of proof is not a proof of absence. Proving that it cannot be done (if it indeed cannot be done) is, presumably, difficult. $\endgroup$– ArthurCommented Aug 25, 2019 at 10:52
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$\begingroup$ Note: in a 1946 article by F. Underwood, nice upper and lower bounds were given for this sum (see here: jstor.org/stable/2306261) $\endgroup$– KlangenCommented Aug 25, 2019 at 10:59
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$\begingroup$ Related on MO: mathoverflow.net/questions/191216/… $\endgroup$– KlangenCommented Aug 25, 2019 at 11:00
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