# How do I change this?

• Jul 23rd 2008, 06:08 PM
rman144
How do I change this?
I know most trig related problems belong in the trig section, but I'm pretty sure this one is more number theory related.

Does anyone know an identity for the following:

(sigma[arctan((2x)/(2n+1))], n=1, n=infinity)-x(zeta(1)). x is greater than one and real.

I've seen a few identities for infinite summation with the inverse tangent, but nothing like this. I know it converges because of the integral test, just not what it converges to.

Any help at all would be much appreciated.
• Jul 23rd 2008, 06:20 PM
Mathstud28
Quote:

Originally Posted by rman144
I know most trig related problems belong in the trig section, but I'm pretty sure this one is more number theory related.

Does anyone know an identity for the following:

(sigma[arctan((2x)/(2n+1))], n=1, n=infinity)-x(zeta(1)). x is greater than one and real.

I've seen a few identities for infinite summation with the inverse tangent, but nothing like this. I know it converges because of the integral test, just not what it converges to.

Any help at all would be much appreciated.

What are you talking about? It doesn't converge except for the trivial solutino x=0?
• Jul 23rd 2008, 06:25 PM
rman144
Rewrite it like this:

sigma[(arctan((2x)/(2n+1)))-(x/n)], n=1, n= infinity.

It converges
• Jul 23rd 2008, 06:30 PM
Mathstud28
Quote:

Originally Posted by rman144
Rewrite it like this:

sigma[(arctan((2x)/(2n+1)))-(x/n)], n=1, n= infinity.

It converges

How did you go from this

http://www.gnux.be/latex/data/d76637...4b8a1f88a2.png

To this

http://www.gnux.be/latex/data/b67a9f...7377258437.png

?
• Jul 23rd 2008, 06:32 PM
rman144
Isn't that the equation for zeta(2)? I placed zeta(1) in the original.
• Jul 23rd 2008, 06:38 PM
Mathstud28
Quote:

Originally Posted by rman144
Isn't that the equation for zeta(2)? I placed zeta(1) in the original.

Indeed you did! I am sorry, I misread it!
• Jul 23rd 2008, 06:39 PM
rman144