Question

The sum of infinite terms of series tan^(-1)((2)/(9))+tan^(-1)((2)/(25))+tan^(-1)((2)/(49))+...oo is equal to cot^(-1)lambda where lambda is a natural number then lambda is equal to​

Answer 1

Answer:

In the given question the rth term can be written as

Tr = tan-1(2r-1/1 + 22r-1)

Now the neumerator can be written as, 2r-1 = 2r-1 x 1 = 2r-1 x (2 -1) = 2r – 2r-1

and the denominator can be written as, 1 + 22r-1 = 1 + 2r.2r-1

let, 2r = x; 2r-1 = y

hence,

Tr = tan-1(x – y/1 + x.y)

which is an inverse trigonometric identity and can be written as

Tr = tan-1(x) – tan-1(y) = tan-1(2r) – tan-1(2r-1)

The given sum is the summation of Tr from r = 0 to r = n

∑Tr = ∑(tan-1(2r) – tan-1(2r-1))

on simplifying we get

∑Tr = tan-12n – tan-120 = tan-12n – tan-11 = tan-12n – π/4