Thread: 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.

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?

Rewrite it like this:

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

It converges

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



To this



?

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

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!

No problem, I do things like that constantly.

I cannot find one, but maybe if someone could use one of these to solve it:

Table of Infinite Products Infinite Sums Infinite Series Elliptic Theta

Found this. Still not the answer, but its so close.

See the attachment.