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
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)
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.
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)