Given that $y = \frac{1}{\sqrt{1 + 2x} + \sqrt{1 + x}}$ where $|x|

Question

Given that $y = \frac{1}{\sqrt{1 + 2x} + \sqrt{1 + x}}$ where $|x| < \frac{1}{2}$, show that, provided $x \neq 0$,$$y = \frac{1}{x} [\sqrt{1 + 2x} - \sqrt{1 + x}].$$ Using this second form of $y$, express $y$ as a series of ascending powers of $x$ up to and including the term in $x^2$. Hence, show that by putting $x = \frac{1}{100}$, $\frac{10}{\sqrt{102} + \sqrt{101}} = \frac{79407}{160000}$.

Accepted Answer

(b) $\frac{1}{2}-\frac{3}{8}x+\frac{7}{16}x^2$.

Estimation using binomial series

Estimation using binomial series