If _ n=1 ²⁰²⁶ ⁻¹ ( 1 n^2+n+1 ) = ⁻¹ (1 - 1 x ) , where x 0 , then x =

Step-by-step solution

The general term of the series is given by: T_n = ⁻¹ ( 1 n^2+n+1 ) This can be rewritten as: T_n = ⁻¹ ( 1 1 + n(n+1) ) = ⁻¹ ( (n+1) - n 1 + (n+1)n ) Using the formula ⁻¹ ( x-y 1+xy ) = ⁻¹(x) - ⁻¹(y) , we get: T_n = ⁻¹(n+1) - ⁻¹(n) Summing from n=1 to n=2026 , we obtain a telescoping series: _ n=1 ²⁰²⁶ T_n = ( ⁻¹(2) - ⁻¹(1)) + ( ⁻¹(3) - ⁻¹(2)) + + ( ⁻¹(2027) - ⁻¹(2026)) _ n=1 ²⁰²⁶ T_n = ⁻¹(2027) - ⁻¹(1) Applying the inverse tangent difference formula again: ⁻¹(2027) - ⁻¹(1) = ⁻¹ ( 2027 - 1 1 + 2027 1 ) = ⁻¹ ( 2026 2