Find the sum of the series :`(tan^- 1)1/3+(tan^- 1)2/9+....+(tan^- 1)(2^(n-1))/(1+2^(2n-1))+...... oo`

Answer

ta`n^{-1} ( 2^{n}-frac{pi}{4})`

Explanation

\begin{aligned}
&\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{2}{9}+\tan ^{-1} \frac{4}{33}+\ldots+\tan ^{-1} \frac{2^{n-1}}{1+2^{2 n-1}} \\ \\
&\Rightarrow \tan ^{-1}\left(\frac{2-1}{1+2 \times 1}\right)+\tan ^{-1}\left(\frac{4-2}{1+4 \times 2}\right)+\tan ^{-1}\left(\frac{8+4}{1+8 \times 4}\right)+\ldots+\tan ^{-1}\left(\frac{2^{n}-2^{n}-1}{1+2^{n} .2^{n-1}}\right) \\ \\
&\Rightarrow \left(\tan ^{-1} 2-\tan ^{-1} 1\right)+\left(\tan ^{-1} 4-\tan ^{-1} 2\right)+\left(\tan ^{-1} 8-\tan ^{-1} 4\right)+\ldots+\left(\tan ^{-1} 2^{n-1}-\tan ^{-1} 2^{n-2}\right)+\left(\tan ^{-1} 2^{n}-\tan ^{-1} 2(n-1)\right) \\
&\Rightarrow \tan ^{-1} 2^{n}-\tan ^{-1} 1 \\
&\Rightarrow \tan ^{-1} 2^{n}-\frac{\pi}{4}
\end{aligned}