# What is the summation formula?

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

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oh sorry! the file needs mathtype to view....sorry!

This file is GIF image format--should be easier to open i hope :surprised
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)

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Last edited:
Gokul43201
Staff Emeritus
Gold Member
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.

Gokul43201
Staff Emeritus
Gold Member
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
I needed to break one GIF file into two...so there are three (sorry)

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Gokul43201
Staff Emeritus
Gold Member
I don't believe there is a general formula for real c. If you use the binomial expansion for $(a+b)^c~,~c~\epsilon~\mathbb{R}$, you will still have terms like $a^c$.

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 :