Chapter 9.9, Problem 24ES

(a) ln 1706 the British astronomer and mathematician John Machin discovered the following for $\pi /4,$ called Machin’s formula:

$\frac{\pi }{4}=4{\tan }^{-1}\frac{1}{5}-{\tan }^{-1}\frac{1}{239}$

Use a CAS to approximate $\pi /4$ using Machin’s formula to 25 decimal places.

(b) In 1914 the brilliant Indian mathematician Srinivasa Ramanujan (1887-1920) showed that

$\frac{1}{\pi }=\frac{\sqrt{8}}{9801}{\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{\left(4k\right)!(1103+26,390k)}{{\left(k!\right)}^4{396}^{4k}}}$

Use a CAS to compute the first four partial sums in Ramanujan’s formula.