Is there a formula to calculate the sum of a number to the power of this same number, like:
$$1^1 + 2^2 + 3^3 + 4^4 + 5^5 + ... + n^n$$?
or
$$x^x + (x+1)^{(x+1)} + (x+2)^{(x+2)} + ... + (x+n)^{(x+n)}$$
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$\begingroup$ trans4mind.com/personal_development/mathematics/series/… $\endgroup$– Don LarynxCommented Nov 6, 2013 at 21:19
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1 Answer
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No. There isn't. But we do know that it is of the order $n^n$, and that all other terms, save the last, can safely be ignored.
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$\begingroup$ Hi there, I know this question is old but I am having troubles to understand your answer (sorry my maths are not the best). What do you mean by "all other terms can safely be ignored"? How can the terms be ignored? The sum would not be correct, right? $\endgroup$ Commented Apr 9, 2015 at 10:08
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$\begingroup$ @AnderBiguri: Take $n=10$, for instance. The last term is $10^{10}$, or ten billion. The sum of all others is about $4\cdot10^8$, or four hundred million. Though great in and of itself, its value is only four percent $(4\%)$ of the former, and the ratio gets lesser and lesser as n grows larger and larger. $\endgroup$– LucianCommented Apr 9, 2015 at 18:57
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$\begingroup$ I see, I know what you meant. It is curious, as I reached here by problem 48 of project euler and I though the OP also did, but it doesn't look like! Google deceives me. Thanks for the explanation. $\endgroup$ Commented Apr 9, 2015 at 19:34