Sum the series: [(sqrt 2) + 1] +1 + [(sqrt 2 -1) + upto - Maths - Sequences and Series

Question

Sum the series: [(sqrt 2) + 1] +1 + [(sqrt 2 -1)
+ upto infinity

Aakash Sharma · Accepted Answer

The given series is a geometrical series with first term $a = \sqrt{2} + 1$ and

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

Hence, the sum S to infinity is given by

$S = \frac{a}{1 - r} = \frac{\sqrt{2} + 1}{1 - (\sqrt{2} - 1)} = \frac{\sqrt{2} + 1}{2 - \sqrt{2}} = \frac{\sqrt{2} + 1}{\sqrt{2} (\sqrt{2} - 1)} = \frac{(\sqrt{2} + 1)^{2}}{\sqrt{2} (\sqrt{2} - 1) (\sqrt{2} + 1)} = \frac{3 + 2 \sqrt{2}}{\sqrt{2}} = \frac{4 + 3 \sqrt{2}}{2}$

Sum the series: [(sqrt 2) + 1] +1 + [(sqrt 2 -1) + ..... upto infinity.