Question

The sum of the series `tan^-1(1/3) + tan^-1(2/9) + ...... + tan^-1[2^(n-1)/(1 + 2^(2n-1))] + ...... ∞` is ..... `(kπ)/4`. Then the value of k is ______.

पर्याय

• 0.00

• 1.00

• 2.00

• 3.00

Answer 1

The sum of the series `tan^-1(1/3) + tan^-1(2/9) + ...... + tan^-1[2^(n-1)/(1 + 2^(2n-1))] + ...... ∞` is ..... `(kπ)/4`. Then the value of k is 1.00.

Explanation:

`sum_(n=1)^∞ tan^-1(2^(n-1)/(1 + 2^(2n-1))) = sum_(n=1)^∞ tan^-1((2^n - 2^(n-1))/(1 + 2^n.2^(n-1)))`

= `sum_(n=1)^∞(tan^-1 2^n - tan^-1 2^(n-1))` 

= tan^–12 – tan^–11 + tan^–14 – tan^–12 + ... + tan^–1∞

= –tan^–11 + tan^–1∞

= `–π/4 + π/2`

= `π/4`

= `1.00 xx π/4`