Converging Series and Derivative Analysis for Positive Terms

[nameVoid]

sum(1/(n(n^2-1)^(1/2)),n=2,infinity)
first derivative <0 for x>=2
I(1/(x(x^2-1)^(1/2)),x,2,infinity)
x=secT, dx=secTtanT
I(secTtanT/(secTtanT),T) ?

 

[Cyosis]

What is your question?

 

[nameVoid]

my book is showing this series converging to pi/6

 

[nameVoid]

ok..this integral is in basic form of I(1/(u(u^2-a^2)^(1/2)),u)=1/a sec^-1(u/a)+C
a=1
sec^-1 u
lim sec^-1x as x-> infinity =pi/2 and sec^-1 2=pi/3
pi/2-pi/3 =pi/6