What is the Sum Formula for Sigma (n^c)?

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Discussion Overview

The discussion revolves around the sum formula for the summation of the form sigma(n^c) for any real constant 'c'. Participants explore the nature of this summation, its potential closed forms, and related mathematical concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about the sum formula for sigma(n^c) from n=1 to k, emphasizing that 'c' is a constant for all terms.
  • Another participant expresses skepticism about the existence of a general closed form for the sum, noting that specific formulas exist for whole number values of 'c'.
  • It is suggested that a polynomial approach could yield a formula for positive integer values of 'c' by determining coefficients through initial values of the sum.
  • Some participants mention computational tools like Mathematica, Maple, and Excel as alternatives for finding the sum.
  • A question arises regarding the formula for (a+b)^c when 'c' is a real number greater than zero, with a focus on the implications of negative bases.
  • Another participant doubts the existence of a general formula for real 'c', referencing the binomial expansion and its limitations.
  • A participant shares a link to an external site claiming to have found a solution for the sum, while also asking about the interpretation of integrals with only a lower limit.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a general formula for the summation of n^c, with some asserting that no such formula exists for real 'c', while others propose methods for specific cases. The discussion remains unresolved regarding a general approach for real constants.

Contextual Notes

Participants note limitations in the discussion, such as the dependence on specific values of 'c' and the challenges of applying binomial expansion to real numbers. There are also references to attachments that may not be universally accessible.

bomba923
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What would be the sum formula for the summation in the attachment?

For any real constant 'c', what is the sum formula for

k
sigma (n^c) ?
n=1
 

Attachments

Physics news on Phys.org
oh sorry! :frown: the file needs mathtype to view...sorry!

This file is GIF image format--should be easier to open i hope
I attached the equation as a GIF image file...i hope it can be open...the GIF file that i attached (the equation is a GIF image file...yeah) (whtisthsum.gif)

(*Note: this is not really a power series--the exponent 'c' is a constant! does not change!...so the sum really goes like
1+(2^c)+(3^c)+(4^c)+(5^c)+(6^c)+...+(k^c)

The exponent c does not change...it is the same for every term of the equation as u add them up...(2^c) and so on to (k^c)..the c exponent does not change, so it's not really a power series)
 

Attachments

  • whtisthsum.gif
    whtisthsum.gif
    548 bytes · Views: 571
Last edited:
I don't believe there's a general closed form for that sum

There are formulas for specific whole number value of c. ie : for c=0,1,2,3 etc.

Also, it's not hard to find a formula, for a general positive integer value of c. This can be done by simply assuming the sum is a polynomial of degree c+1, and determining the coefficients by plugging in the first c+2 values of the sum.

Need to think more about a general method for real c.
 
Of course, that's other than using Mathematica, or Maple. In fact, this can be done quite easily using Excel, too.
 
Wait, but what would the formula be for any real c>0 ?

I tried solving it, but is there a formula for (a+b)^c , where c>0 but where 'c' could be real?...(not just natural). Let's just take the case where (a+b)>0 , because u cannot have a real root for an irrational power of a negative number...

So is there a formula for (a+b)^c where c is real and c>0 and (a+b)>0?

Look at the attachments...um, i posted three sorry :frown:
I needed to break one GIF file into two...so there are three (sorry)
 

Attachments

  • whtisthsum.gif
    whtisthsum.gif
    548 bytes · Views: 526
  • whtbnsum.gif
    whtbnsum.gif
    1.1 KB · Views: 588
  • whtbnsum2.gif
    whtbnsum2.gif
    1.3 KB · Views: 598
I don't believe there is a general formula for real c. If you use the binomial expansion for [itex](a+b)^c~,~c~\epsilon~\mathbb{R}[/itex], you will still have terms like [itex]a^c[/itex].

You don't have to be including attachments for math representations. You can simply use LaTeX typesetting, as I've done. Look at this thread for LaTeX :
https://www.physicsforums.com/showthread.php?t=8997
 
Hmm...i've found a solution elsewhere on some polysum tripod site:

< http://polysum.tripod.com/ >

What does it mean when an integrand is written only with a lower limit without an upper one?
Does that mean it applies from that lower limit to infinity? or something else?
 

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