Chapter 5.6, Problem 1SP

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.

1. The series

$\pi =4{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^{n+1}}{2n-1}}=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\mathrm{...}$

was discovered by Gregory and Leibniz in the late 1600s. This result follows from the Maclaurin series for f(x) = tan-1 x. We will discuss this series in the next chapter.

a. Prove that this series converges.

b. Evaluate the partial sums S_nfor n = 10. 20. 50. 100.

C. Use the iemainder estimate for alternating series to get a bound on the error R_n.

d. What is the smallest value of N that guarantees |R_N| <0.01? Evaluate S_N.