Chapter 5.6, Problem 2SP

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.

2. The series

$\pi =6{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\begin{array}{l}\frac{\left(2n\right)!}{2^{4n+1}\left(n!\right)\left(2n+1\right)}\\ =6\left(\frac{1}{2}+\frac{1}{2.3}{\left(\frac{1}{2}\right)}^3+\frac{1.3}{\mathrm{2.4.5}}\right).{\left(\frac{1}{2}\right)}^5+\frac{\mathrm{1.3.5}}{\mathrm{2.4.6.7}}\left(\frac{1}{2}\right)+\mathrm{...})\end{array}}$

has been attributd to Newton in the late 16OO. The proof of this result uses the Madautin series for $f\left(x\right)={\sin }^{-1}x$ .

a. Prove that the series converges.

b. Evaluate the partial sums S for n = 5. 10. 20.

C. Compare S to z for n = 5. 10. 20 and discuss the number of correct decimal places.