- #1

quantumdude

Staff Emeritus

Science Advisor

Gold Member

- 5,575

- 23

This was a thread that he started in PF v2.0, and which I participated in. I made a copy of it because I left it unfinished.

Prove:

Σ

_{0}

^{oo}arctan(1/(n

^{2}+n+1))=π/2

My (as yet incomplete) solution:

Parametrize the sum as follows:

S(a)=Σ

_{0}

^{oo}arctan(a/(n

^{2}+n+1))

S'(a)=Σ

_{0}

^{oo}(n

^{2}+n+1)/(a

^{2}+(n

^{2}+n+1)

^{2})

That gets rid of the nasty arctan function.

My approach will be to find the sum of S'(a) and integrate with respect to 'a' with the limits 0<a<1. That will give me:

S(1)-S(0)=S(1), which is the original sum.

To be continued...