Question

Sum the series to infinity :(√2+1)+1(√2-1)+.....

Answer 1

Step-by-step explanation:

it is a geometric progression with first term

(√2+1) and common ratio (√2-1)/(√2+1) =>

(√2-1)(√2+1)/(√2+1)(√2+1) =>

$\frac{1}{ {( \sqrt{2} + 1)}^{2} }$

Hence the sum of this series to infinity is

first term/(1- common ratio) =>

$\frac{( \sqrt{2} + 1)}{1 - \frac{1}{ {( \sqrt{2} + 1)}^{2} } } = > \frac{ {( \sqrt{2} + 1) }^{3} }{ {( \sqrt{2} + 1)}^{2} - 1 } = > \frac{ {( \sqrt{2} + 1) }^{3} }{2 + 1 + 2 \sqrt{2} - 1}$

$= > \frac{ {( \sqrt{2} + 1)}^{3} }{2(1 + \sqrt{2})} = > \frac{ {( \sqrt{2} + 1) }^{2} }{2} = > \frac{2 + 1 + 2 \sqrt{2} }{2}$

$= > 1.5 + \sqrt{2}$

Hope it helps you.