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JEE Main

Exam: JEE Main
Subject: Mathematics
Chapter: Basics of Mathematics

JEE Main · Mathematics · Basics of Mathematics

(1)/(2 · 3)+(1)/(4 · 5)+(1)/(6 · 7)+(1)/(8 · 9)+ is equal to

\frac{1}{2 \cdot 3}+\frac{1}{4 \cdot 5}+\frac{1}{6 \cdot 7}+\frac{1}{8 \cdot 9}+\ldots

is equal to

Options

A $\log \left(\frac{2}{e}\right)$
B $\log \left(\frac{e}{2}\right)$
C $\log (2 e)$
D $e-1$

Correct answer

B. $\log \left(\frac{e}{2}\right)$

Step-by-step solution

\begin{aligned} & \frac{1}{2 \cdot 3}+\frac{1}{4 \cdot 5}+\frac{1}{6 \cdot 7}+\frac{1}{8 \cdot 9}+\ldots \infty \\ & =\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+\left(\frac{1}{6}-\frac{1}{7}\right)+\left(\frac{1}{8}-\frac{1}{9}\right)+\ldots \infty \\ & =1-\log _e 2 \\ & =\log _e e-\log _e 2 \\ & =\log \left(\frac{e}{2}\right)\end{aligned}

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