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Sum of Infinite Terms of a GP

For the follo...

Question

For the following sequence in G.P., find the sum of infinite terms :
$\frac{\sqrt{2} + 1}{\sqrt{2} - 1} + \frac{1}{2 - \sqrt{2}} + \frac{1}{2} + . . . .$

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Solution

$S_{\infty} = \frac{a}{1 - r}$

where $r = \frac{\frac{1}{2 - \sqrt{2}}}{\frac{\sqrt{2} + 1}{\sqrt{2} - 1}}$

$= \frac{\frac{\sqrt{2} + 1}{\sqrt{2} - 1}}{1 - (\frac{\sqrt{2} - 1}{\sqrt{2}})}$

$= \sqrt{2} (\frac{\sqrt{2} + 1}{\sqrt{2} - 1})$

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