Question

The sum of the series $\frac{1}{1\cdot 2}-\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}-.....\infty$ is

• A. $2{\log }_{e}2+1$
• B. $2{\log }_{e}2$
• C. $2{\log }_{e}2-1$
• D. ${\log }_{e}2-1$

Answer 1

Answer: (C)

$2{\log }_{e}2-1$

Let $S=\frac{1}{1\cdot 2}-\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}-.....\infty$
$=(\frac{1}{1}-\frac{1}{2})-(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})-....$
$=\frac{1}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-....$
$=2(\frac{1}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-....)-1=2{\log }_{e}2-1$