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  4. The sum of the infinite series (1/2)+(1/3)-(1/4)-(1/9)+(1/8)+(1/27)-(1/16)-(1/81)+ ldots ldots ∞, is

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

Sequences and Series

Solution:

$S = S _1+ S _2$;
$S _1=\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+\ldots \ldots=\frac{\frac{1}{2}}{1+\frac{1}{2}}=\frac{1}{2} \cdot \frac{2}{3}=\frac{1}{3} $
$S _2=\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\ldots \ldots=\frac{\frac{1}{3}}{1+\frac{1}{3}}=\frac{1}{3} \cdot \frac{3}{4}=\frac{1}{4}$
$S =\frac{1}{3}+\frac{1}{4}=\frac{7}{12}$