Chapter 11.6, Problem 50E

Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula

$\frac{1}{\pi }=\frac{2\sqrt{2}}{9801}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{\left(4n\right)!\left(1103+26390n\right)}{{\left(n!\right)}^4{396}^{4n}}}$

William Gosper used this series in 1985 to compute the first 17 million digits of $\pi$.

(a) Verify that the series is convergent.

(b) How many correct decimal places of $\pi$ do you get if you use just the first term of the series? What if you use two terms?