Summation of 1^1+2^2+3^3+....+k^k

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In summary, the conversation discusses the possibility of finding a shorter representation for the expression \sum_{n=1}^{k} n^n. The formula is not of the form of a constant power series and cannot be easily approximated. One suggestion is to use Faulhaber's formula to rewrite it as a difference of sums, but it may still result in a long expression. It is mentioned that for engineers, the expression may be shortened for larger values of k, but it is still unlikely to be significantly shortened.
  • #1
ddddd28
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Does that summatiom have a shorter representation at all?
##\sum_{n=1}^{k} n^n = ?##
I guess it is not of the form of constant power series, but I could not find an alternative.

Mentor note: made formula render properly
 
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  • #3
Huh? Looks both like nn to me.

I'm not aware of an analytic expression. It can probably be approximated with the Stirling formula and then some integration.
 
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  • #4
It's kk :-)
 
  • #5
Maybe one can use Faulhaber to rewrite ##n^n## as difference of ##\sum_{k=1}^n k^n - \sum_{k=1}^{n-1} k^n## to get an expression in Bernoulli numbers which can then be summed again. A giant polynomial of Bernoulli numbers. Of course my bet to the original question
ddddd28 said:
Does that summatiom have a shorter representation at all?
is NO. I mean the length of the expression is seven! Almost impossible to shorten.
 
  • #6
fresh_42 said:
is NO. I mean the length of the expression is seven! Almost impossible to shorten.
Not true for an engineer for k>5 or so...

k^k
 
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  • #7
berkeman said:
Not true for an engineer for k>5 or so...

k^k
Now as you say it. Mathematicians can also shorter ...
##O(1)##
 
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FAQ: Summation of 1^1+2^2+3^3+....+k^k

What is the formula for the summation of 1^1+2^2+3^3+....+k^k?

The formula for this summation is k^(k+1)/(k+1) - 1.

How is this summation related to exponential growth?

This summation is related to exponential growth because each term in the sequence is increasing at an exponential rate.

What is the significance of the exponents being equal to the base in this summation?

The significance of the exponents being equal to the base is that it creates a self-referential relationship where each term is dependent on the previous one, resulting in a rapidly increasing sequence.

Can this summation be simplified or approximated?

Yes, the summation can be simplified or approximated using different methods such as the trapezoidal rule or the Euler-Maclaurin formula.

In what fields or applications is this summation commonly used?

This summation is commonly used in mathematics and physics, particularly in the study of series and sequences, and in the analysis of exponential growth and decay phenomena.

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