Chapter 5.6, Problem 3SP

Series Converging to $\pi$and 1/ $\pi$Dozens of series exist that converge to $\pi$ or an algebraic expression containing $\pi$ . Here we look at several examples and compare their rates of convergence. By rate of conveigence. we mean the number of terms necessazy for a partial sum to be within a certain amount of the actual value. The series representations of z in the first two examples can be explained using Maclaunn series, which axe discussed in the next chapter. The third example relies on material beyond the scope of this text.

3. The series

   $\frac{1}{\pi }=\frac{\sqrt{8}}{9801}{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{\left(4n\right)!\left(1103+26390n\right)}{{\left(n!\right)}^4{396}^{4n}}}$

was discovered by Ramanujan in the early 1900s. William Gosper. Jr., used this series to calculate x to an accuracy of more than 17 million digits in the niid-l9)s. At the time, that was a world record. Since that time, this series and others by Rarnanujan have led mathematicians to find many other series represetnations for x and I/x.

a. Prove that this series converges.

b. Evaluate the first term in this series. Compare this number with the value of x from a calculating utilky. To how many decimal places do these two numbers agree? What if we add the first two teiTns in the series?

C. Investigate the life of Szinivasa Ram.anujan (1887—1920) and write a brief summary. Ramanujan is one of the most fascinating stories in the histo, of mathematics. He was basically sell-taught, with no formal training in mathematics. e he contributed in highly original ways to many advanced areas of mathemarics.