Question

If `lim_(n→∞)sum_(k = 2)^ncos^-1(1 + sqrt((k - 1)(k + 2)(k + 1)k)/(k(k + 1))) = π/λ`, then the value of λ is ______.

Options

• 5.00

• 6.00

• 7.00

• 8.00

Answer 1

If `lim_(n→∞)sum_(k = 2)^ncos^-1(1 + sqrt((k - 1)(k + 2)(k + 1)k)/(k(k + 1))) = π/λ`, then the value of λ is 6.00.

Explanation:

`sum_(k = 2)^ncos^-1(1/k. 1/(k + 1) + sqrt(1 - 1/k^2)sqrt(1 - 1/(k + 1)^2))`

`sum_(k = 2)^n(cos^-1  1/(k + 1) - cos^-1  1/k) = (cos^-1  1/3 - cos^-1  1/2) + (cos^-1  1/4 - cos^-1  1/3) + .........(cos^-1  1/(n + 1) - cos^-1  1/n)`

= `cos^-1  1/(n + 1) - cos^-1  1/2`

`lim_(n→∞)(cos^-1  1/(n + 1) - cos^-1  1/2) = π/2 - π/3 = π/6` the value of λ is 6.