JEE MAIN Logarithmic Series Previous Year Questions (PYQs)
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If 0 < a, b < 1, and tan$-$1a + tan$-$1b = ${\pi \over 4}$, then the value of
$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....$ is :
$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....$ is :
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Solution
Let ${S_k} = \sum\limits_{r = 1}^k {{{\tan }^{ - 1}}\left( {{{{6^r}} \over {{2^{2r + 1}} + {3^{2r + 1}}}}} \right)} $. Then $\mathop {\lim }\limits_{k \to \infty } {S_k}$ is equal to :
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Solution
If the solution of the equation $\log_{\cos x}\!\big(\cot x\big) + 4\log_{\sin x}\!\big(\tan x\big) = 1,\ x\in\left(0,\tfrac{\pi}{2}\right),$ is $\sin^{-1}\!\left(\tfrac{\alpha+\sqrt{\beta}}{2}\right)$, where $\alpha,\beta$ are integers, then $\alpha+\beta$ is equal to:
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Solution
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