Question

Find the sum of

sin^-1(1/√2) + sin^-1((√2 - 1)/√6) + sin^-1((√3 - √2)/√12) + ....+ sin^-1((√n - √(n - 1))/(√n x √(n + 1))) + ... to ∞

Answer 1

Let I t_n = sin^-1((√n - √(n - 1))/(√n x √(n + 1)))

when n → ∞, the sum is tan^-1(∞) = π/2.