- Show that the first six terms of the binomial series for
are
![\[ 1 + \frac{1}{2} x + \frac{3}{8} x^2 + \frac{5}{16}x^3 + \frac{35}{128} x^4 + \frac{63}{256} x^5. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-ed2327398bf13df9d444f20b4080804e_l3.png "Rendered by QuickLaTeX.com")
- For
let 
be the 
th coefficient in the binomial series and let 
denote the remainder after terms, i.e.,
![\[ r_n = a_{n+1} + a_{n+2} + a_{n+3} + \cdots \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-37252b2a98717c183c36ba068d69cc3c_l3.png "Rendered by QuickLaTeX.com")
for
. Prove that![\[ 0 < r_n < \frac{a_n}{49}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-78028f43b54992f6ab018c99c4ac7359_l3.png "Rendered by QuickLaTeX.com")
- Prove the validity of the identity
![\[ \sqrt{2} = \frac{7}{5} \left( 1 - \frac{1}{50} \right)^{-\frac{1}{2}}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-55ae09e2ebf5232697261994e1777ce3_l3.png "Rendered by QuickLaTeX.com")
Use this identity to compute the first ten decimal places of
.
Incomplete.
From section 11.15, we know binomial formula for exponent -1/2. From that we get a_n = [1*3*5*…*(2n-1)]/[2^n*n!]*(1/50)^n, and then by using hints one can solve this.