How do I evaluate the following finite sum over $k$
$1+\frac{1}{2^n}+\frac{1}{2^n3^n}+\frac{1}{2^n3^n4^n}+\cdots+\frac{1}{2^n3^n\cdots k^n}$
or if there is an expression of this sum in terms of other known numbers ??
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1$\begingroup$ $\sum_{k=0}^\infty(k!)^{-n}$ is open at math.stackexchange.com/q/2902724. Or do you mean the finite sum? $\endgroup$– Francois ZieglerCommented Aug 8, 2019 at 14:20
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$\begingroup$ the sum is finite over $k$ $\endgroup$– wkmCommented Aug 8, 2019 at 14:28
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1 Answer
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$${}_1F_n(1;2,2,\ldots 2; 1) - \frac{{}_1F_{n}(1;2+k,2+k,\ldots 2+k; 1)}{((k+1)!)^n} $$
answered Aug 8, 2019 at 14:39
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$\begingroup$ well thanks, can I express it in terms of generalized harmonic numbers or multiple harmonic sums ?? $\endgroup$– wkmCommented Aug 8, 2019 at 15:58
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$\begingroup$ What is denoted by $_1F_n$? How is that obtained? $\endgroup$ Commented Aug 8, 2019 at 16:31
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$\begingroup$ You can see the definition in wikipedia en.wikipedia.org/wiki/Hypergeometric_function $\endgroup$– wkmCommented Aug 8, 2019 at 16:37