The value of the expression ^ -1 (1)/(2) + ^ -1 and (9)/(2) + ^ -1...

Given expression& $= \tan^{-1 } 2 + \tan^{-1 } \dfrac{2}{9} + \tan^{-1 } \dfrac{2}{2 5} + \tan^{-1 }\dfrac{2}{4 9} + \text{........}$

$\text{General term} = \dfrac{2}{\left(2n - 1\right)^{2}} = \dfrac{2}{4n^{2} - 4n + 1} = \dfrac{2}{1 + 4n\left(n - 1\right)} = \dfrac{2n - \left(2n - 2\right)}{1 + 2n \left(2n - 2\right)}$

${T}_{n} = \tan^{-1} 2n - \tan^{-1} \left(2n - 2\right)$

$∴$ & Sum of the series $= \tan^{-1} 2 - \tan^{-1} 0 + \tan^{-1} 4 - \tan^{-1} 2 + \tan^{-1} 6 - \tan^{-1} 4 + \ldots \tan^{-1} 2n - \tan^{-1} \left(2n - 2\right)$

$= \tan^{-1} 2n - \tan^{-1} 0 = \tan^{-1} 2n$