The sum of the infinite series cot^(-1)(7/4)+cot^(-1)(19/4)+cot^(-1)(39/4)+cot^(-1)(67/4)+..... is :-

Question

The sum of the infinite series \(\cot ^{-1}\left(\frac{7}{4}\right)+\cot ^{-1}\left(\frac{19}{4}\right)+\cot ^{-1}\left(\frac{39}{4}\right)+\cot ^{-1}\left(\frac{67}{4}\right)+\ldots ..\) is :-

(1) \(\frac{\pi}{2}+\tan ^{-1}\left(\frac{1}{2}\right)\)

(2) \(\frac{\pi}{2}-\cot ^{-1}\left(\frac{1}{2}\right)\)

(3) \(\frac{\pi}{2}+\cot ^{-1}\left(\frac{1}{2}\right)\)

(4) \(\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{2}\right)\)

Answer 1

Correct option is: (4) \(\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{2}\right)\)  

\( \mathrm{T}_{\mathrm{n}}=\tan ^{-1}\left(\frac{4}{4 \mathrm{n}^{2}+3}\right)\)

\(T_{n}=\tan ^{-1}\left(\frac{\left(n+\frac{1}{2}\right)-\left(n-\frac{1}{2}\right)}{1+\left(n+\frac{1}{2}\right)\left(n-\frac{1}{2}\right)}\right)\)

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

\(\mathrm{T}_{1}+\mathrm{T}_{2}+\ldots+\mathrm{T}_{\mathrm{n}}=\tan ^{-1}\left(\mathrm{n}+\frac{1}{2}\right)-\tan ^{-1}\left(\frac{1}{2}\right)\)

\(\mathrm{S}_{\infty}=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1}{2}\right)\)