The sum to infinite terms of the series ⁻¹ ( 1 3 )+ ⁻¹ ( 2 9 )+ + ⁻¹ ( 2^ n -1 1+2^ 2 n -1 )+ is

Step-by-step solution

The general term T_n = ⁻¹ ( 2^ n-1 1+2^ 2n-1 ) can be expressed as a difference of arctangents using the identity ⁻¹x - ⁻¹y = ⁻¹ ( x-y 1+xy ) . Setting x = 2^n and y = 2^ n-1 gives x-y = 2^ n-1 and xy = 2^ 2n-1 , yielding T_n = ⁻¹(2^n) - ⁻¹(2^ n-1 ) . The partial sum S_N = _ n=1 ^ N T_n telescopes to ⁻¹(2^N) - ⁻¹(1) . Taking the limit as N , S_ = _ N ⁻¹(2^N) - 4 = 2 - 4 = 4 . Answer: 4