shocked by a Ramanujan identity

**zetafunction** · Apr14-09, 04:38 PM

i do not remember the webpage i watched this but i remember that they said ' IN chapter 1 of his notebook Ramanujan wrote '

$LaTeX Code: \\sum_{n=0}^{x}n^{r}= (r+1)^{-1}x^{r+1}+ \\zeta (-r) - \\sum_{k}B_{2k}\\frac{\\Gamma (r+1)}{\\Gamma (k-2r+2)}$

does anyone knows how to get this ??

**ramsey2879** · Apr14-09, 09:07 PM

Originally Posted by zetafunction

i do not remember the webpage i watched this but i remember that they said ' IN chapter 1 of his notebook Ramanujan wrote '

$LaTeX Code: \\sum_{n=0}^{x}n^{r}= (r+1)^{-1}x^{r+1}+ \\zeta (-r) - \\sum_{k}B_{2k}\\frac{\\Gamma (r+1)}{\\Gamma (k-2r+2)}$

does anyone knows how to get this ??

To begin with, what is B? and in the sum what is the range of k?

**CRGreathouse** · Apr14-09, 10:42 PM

Originally Posted by ramsey2879

To begin with, what is B? and in the sum what is the range of k?

A Bernoulli number and "0 to infinity", I believe.

**zetafunction** · Apr15-09, 07:35 AM

oh, excuse me the lack of notation

here B_2k means the 2k-th Bernoulli Number, and yes the summation (is finite) goes to k=0 to k= infinite

i know he used Euler-Maclaurin summation formula but the term $LaTeX Code: \\zeta (-r)$ seems strange to me

**damo_clark** · Apr18-09, 02:54 AM

The "Euler Summation Formula" might give you some insight to why the Bernouli numbers pop up in this formula. The term $LaTeX Code: \\zeta (-r)$ is just $LaTeX Code: \\frac{B_{r+1}}{r+1}$ for a positive integer value of r. Another formula that will compute this sum for a positive integer value of r is :

$LaTeX Code: f_{r}(x)= \\sum_{n=0} ^{x} n^{r} = r\\int f_{r-1}dx + xB_{r}$

where

$LaTeX Code: f_{1}(x)=\\frac{x^2}{2} + \\frac{x}{2}$

I think one of the Bernoulli brothers could calculate $LaTeX Code: 1^{10}+2^{10}+3^{10} + .... 1000^{10}$ in about fifteen minutes using one of these formulas. Try it using just pen and paper and see how long it takes you...