Is there any general formula for the sequence $S_n=\sum_{k=1}^n k^k$?

Asked May 11 '16 at 01:02

Active Jan 02 '18 at 01:12

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I have faced this series several times. Could it be possible to come up with a formula for the following sequence?

$S_n= \sum_{k=1}^n k^k = 1^1 + 2^2 + 3^3 + \ldots + n^n$

If yes, then what is that and how it could be proved?

Note: I had come through this question which I have found almost near to this series but still could not come up.

edited Jan 02 '18 at 01:12

Simply Beautiful Art

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asked May 11 '16 at 01:02

Enamul Hassan

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The word series usually means the infinite sum, you should just use sum (or summation) here . – YoTengoUnLCD May 11 '16 at 01:06
@YoTengoUnLCD Could I use the word sequence? because, pluses are already indicating that it is a summation. – Enamul Hassan May 11 '16 at 01:08
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First rule of 'is there a formula for this sequence?' : try OEIS. In this case, plugging the first half-dozen values in yields http://oeis.org/A001923 , which doesn't have any information about a simpler closed form, making it highly unlikely that one is known. – Steven Stadnicki May 11 '16 at 01:11
@StevenStadnicki Thats a good point. But could we tell that there does not exist a formula for the sequence? Thanks – Enamul Hassan May 11 '16 at 01:14
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Yes, sequence is fine (i.e, "the sequence $s_n=\sum_{k=1}^n k^k$"). – YoTengoUnLCD May 11 '16 at 01:15
@YoTengoUnLCD Thanks, updated. – Enamul Hassan May 11 '16 at 01:16
@manetsus There are almost no tools for being able to say 'a formula does not exist for this sequence' - sometimes you can use growth rates to eliminate formulae of a specific kind, but in a case like this that sort of approach isn't really available. – Steven Stadnicki May 11 '16 at 01:43
@StevenStadnicki Well, I guess we could just say "this sequence grows as $x^x$" or something like that, to give an estimate. – YoTengoUnLCD May 11 '16 at 17:26
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@DietrichBurde $k!\ne k^k$, why do you keep trying to close these as dupes? – Simply Beautiful Art Jan 02 '18 at 01:11
@SimplyBeautifulArt I am sorry, retracted! Took the wrong cut and paste. I meant this one. – Dietrich Burde Jan 02 '18 at 09:55
Another one is here. – Dietrich Burde Jan 02 '18 at 10:06

Is there any general formula for the sequence $S_n=\sum_{k=1}^n k^k$?

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