Compute some things using the binomial series

by RoRi

April 24, 2016

Show that the first six terms of the binomial series for $(1 - x)^{-\frac{1}{2}}$ are
$1 + \frac{1}{2} x + \frac{3}{8} x^2 + \frac{5}{16}x^3 + \frac{35}{128} x^4 + \frac{63}{256} x^5.$
For $x = \frac{1}{50}$ let $a_n$ be the $n$ th coefficient in the binomial series and let $r_n$ denote the remainder after $n$ terms, i.e.,
$r_n = a_{n+1} + a_{n+2} + a_{n+3} + \cdots$

for $n \geq 0$ . Prove that

$0 < r_n < \frac{a_n}{49}.$
Prove the validity of the identity
$\sqrt{2} = \frac{7}{5} \left( 1 - \frac{1}{50} \right)^{-\frac{1}{2}}.$

Use this identity to compute the first ten decimal places of $\sqrt{2}$ .