Find the sum of each of the following infinite series :

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 $=\frac{2 \sqrt{2}}{3}$

 

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