Question

find the sum
\[\sum_{n=1}^{\infty} (\tan ^{-1}(n)- \tan ^{-1}(n+1))\]

Answer 1

i know it converges because lim as n->infinity is 0 , also the sum has to be negative because arctan(n) < arctan(n+1)

Answer 2

sorry not sure what to do after that other than approximate numerically

Answer 3

for what its worth, i ran sum up to n=1000 and got -0.7844 so it seems to be approaching pi/4