The sum _ n = 1 ^ ^ - 1 ( 2 ( _ k = 1 ^n k ) - 1 3 ) is equal to - Vidyadip

MCQ

The sum $\sum\limits_{n = 1}^\infty {{{\cot }^{ - 1}}} \left( {\frac{{2\left( {\sum\limits_{k = 1}^n k } \right) - 1}}{3}} \right)$ is equal to

✓
$\frac{{3\pi }}{4} + {\cot ^{ - 1}}2$
B
$\frac{\pi }{2} + {\cot ^{ - 1}}3$
C
$\pi $
D
$\frac{\pi }{2} + {\tan ^{ - 1}}2$

✓

Answer

Correct option: A.

$\frac{{3\pi }}{4} + {\cot ^{ - 1}}2$

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

$=\tan ^{-1}(\mathrm{n}+2)-\tan ^{-1}(\mathrm{n}-1)$

Use $: \mathrm{S}_{\mathrm{n}}=\Sigma \mathrm{T}_{\mathrm{n}}$

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