Question

$$\text{ The sum of the series }\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}-\dots ..\mathrm{\infty }$$

• (A) 2loge 2 + 1
• (B) 2loge 2
• (C) 2loge 2 – 1
• (D) loge 2 – 1

Answer 1

The Correct Option is C

Explanation:
$$\begin{array}{rl}& s=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{3.4}\dots \dots \mathrm{\infty }\\ =& (\frac{1}{1}-\frac{1}{2})-(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})\\ =& \frac{1}{1}-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}+\frac{1}{5}\dots \\ =& \frac{1}{1}-\frac{2}{2}+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-\dots \\ =& 2[\frac{1}{1}-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}\dots ]-1=2\log 2-1\end{array}$$