A) \[\log \frac{e}{2}\]
B) \[\log \frac{e}{4}\]
C) \[\log \frac{2}{3}\]
D) \[\log \frac{2}{4}\]
Correct Answer: A
Solution :
\[\frac{1}{2\cdot 3}+\frac{1}{4\cdot 5}+\frac{1}{6\cdot 7}+...\] \[=\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\frac{1}{7}+...\] \[=1-1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+...\] \[=1-\left( 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+... \right)\] \[=1-\log 2=\log e-\log 2\] \[=\log \frac{e}{2}\]
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