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Compute some things using the binomial series

  1. Show that the first six terms of the binomial series for (1 - x)^{-\frac{1}{2}} are

        \[ 1 + \frac{1}{2} x + \frac{3}{8} x^2 + \frac{5}{16}x^3 + \frac{35}{128} x^4 + \frac{63}{256} x^5. \]

  2. For x = \frac{1}{50} let a_n be the nth 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

        \[ 0 < r_n < \frac{a_n}{49}. \]

  3. 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}.


Incomplete.

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