Question:
Find the sum of each of the following infinite series :
$\sqrt{2}-\frac{1}{\sqrt{2}}+\frac{1}{2 \sqrt{2}}-\frac{1}{4 \sqrt{2}}+\ldots . .$ ∞
Solution:
It is Infinite Geometric Series
Here, $a=\sqrt{2}$
$r=\frac{\frac{-1}{\sqrt{2}}}{\sqrt{2}}=\frac{-1}{2}$
$\therefore S u m=\frac{a}{1-r}=\frac{\sqrt{2}}{1-\frac{-1}{2}}=\frac{\sqrt{2}}{1+\frac{1}{2}}=\frac{2 \sqrt{2}}{3}$
Sum $=\frac{2 \sqrt{2}}{3}$