Evaluate tan^(-1) 2+tan^(-1) 2/9+ tan ^(-1) 2/25 tan^(-1)2/49+....upto n terms

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$$\begin{gathered} Let S={\tan }^{-1}\left(2\right)+{\tan }^{-1}\left(\frac{2}{9}\right)+{\tan }^{-1}\left(\frac{2}{25}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} ;\mathrm{equation}\left(\mathrm{i}\right) \\ \mathrm{We} \mathrm{know} \mathrm{that} {\tan }^{-1}\left(x\right)+{\tan }^{-1}\left(y\right)={\tan }^{-1}\left(\frac{x+y}{1-xy}\right) ; x,y>0 \mathrm{and} xy<1 \\ \mathrm{Using} \mathrm{this} \mathrm{identity} \mathrm{in} \mathrm{equation} \left(\mathrm{i}\right) , \mathrm{we} \mathrm{get} \\ S={\tan }^{-1}\left(\frac{2+{\displaystyle \frac{2}{9}}}{1-2\times {\displaystyle \frac{2}{9}}}\right)+{\tan }^{-1}\left(\frac{2}{25}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ \Rightarrow S={\tan }^{-1}\left(\frac{{\displaystyle \frac{18+2}{9}}}{{\displaystyle \frac{9-4}{9}}}\right)+{\tan }^{-1}\left(\frac{2}{25}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ \Rightarrow S={\tan }^{-1}\left(\frac{20}{5}\right)+{\tan }^{-1}\left(\frac{2}{25}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ \Rightarrow S={\tan }^{-1}\left(4\right)+{\tan }^{-1}\left(\frac{2}{25}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ \Rightarrow S={\tan }^{-1}\left(\frac{4+{\displaystyle \frac{2}{25}}}{1-4\times {\displaystyle \frac{2}{25}}}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ S={\tan }^{-1}\left(\frac{{\displaystyle \frac{100+2}{25}}}{{\displaystyle \frac{25-8}{25}}}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ S={\tan }^{-1}\left(\frac{102}{17}\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ S={\tan }^{-1}\left(6\right)+{\tan }^{-1}\left(\frac{2}{49}\right)+········\mathrm{upto} \mathrm{n} \mathrm{terms} \\ \mathrm{Now} \mathrm{if} \mathrm{you} \mathrm{observe} \\ \mathrm{When} \mathrm{we} \mathrm{take} \mathrm{first} \mathrm{term} \mathrm{of} \mathrm{S} \mathrm{we} \mathrm{get} {\tan }^{-1}\left(2\right)={\tan }^{-1}\left(2\times \mathbf{1}\right) \\ \mathrm{When} \mathrm{we} \mathrm{add} \mathrm{first} \mathrm{two} \mathrm{terms} \mathrm{of} \mathrm{S} \mathrm{we} \mathrm{get} {\tan }^{-1}\left(4\right)={\tan }^{-1}\left(2\times \mathbf{2}\right) \\ \mathrm{When} \mathrm{we} \mathrm{add} \mathrm{first} \mathrm{three} \mathrm{terms} \mathrm{of} \mathrm{S} \mathrm{we} \mathrm{get} {\tan }^{-1}\left(6\right)={\tan }^{-1}\left(2\times \mathbf{3}\right) \\ \mathrm{Similarly} \\ \mathrm{When} \mathrm{we} \mathrm{add} \mathrm{n} \mathrm{terms} \mathrm{of} \mathrm{S} \mathrm{we} \mathrm{will} \mathrm{get} {\tan }^{-1}\left(2\times \mathit{n}\right)={\tan }^{-1}\left(2n\right) \\ \mathbf{\Rightarrow }\mathit{S}\mathbf{=}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{2}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\frac{\mathbf{2}}{\mathbf{9}}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\frac{\mathbf{2}}{\mathbf{25}}\right)\mathbf{+}{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\frac{\mathbf{2}}{\mathbf{49}}\right)\mathbf{+}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{·}\mathbf{upto}\mathbf{ }\mathbf{n}\mathbf{ }\mathbf{terms}\mathbf{=}\mathbf{ }\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{2}\mathbf{n}\right) \end{gathered}$$

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