Question

The sum of the infinite series $\frac{1}{2}+\frac{1}{3}-\frac{1}{4}-\frac{1}{9}+\frac{1}{8}+\frac{1}{27}-\frac{1}{16}-\frac{1}{81}+\dots \dots \infty$, is

• A. $\frac{1}{6}$
• B. $\frac{5}{6}$
• C. $\frac{5}{12}$
• D. $\frac{7}{12}$

Answer 1

Answer: (D)

$S=S_1+S_2$;
$S_1=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\dots \dots =\frac{\frac{1}{2}}{1+\frac{1}{2}}=\frac{1}{2}\cdot \frac{2}{3}=\frac{1}{3}$
$S_2=\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\dots \dots =\frac{\frac{1}{3}}{1+\frac{1}{3}}=\frac{1}{3}\cdot \frac{3}{4}=\frac{1}{4}$
$S=\frac{1}{3}+\frac{1}{4}=\frac{7}{12}$