5.6 Ratio and root tests  (Page 4/8)

Series converging to $\pi$ And $1\text{/}\pi$

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 convergence, we mean the number of terms necessary for a partial sum to be within a certain amount of the actual value. The series representations of $\pi$ in the first two examples can be explained using Maclaurin series, which are discussed in the next chapter. The third example relies on material beyond the scope of this text.

1. The series
   
   $\pi =4{\displaystyle \sum }_{n=1}^{\infty }\frac{{\left(\mathrm{-1}\right)}^{n+1}}{2n-1}=4-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\text{⋯}$
   
   was discovered by Gregory and Leibniz in the late $1600\text{s}\text{.}$ This result follows from the Maclaurin series for $f\left(x\right)={\text{tan}}^{\mathrm{-1}}x.$ We will discuss this series in the next chapter.
   
   1. Prove that this series converges.
   2. Evaluate the partial sums $S_n$ for $n=10,20,50,100.$
   3. Use the remainder estimate for alternating series to get a bound on the error $R_n.$
   4. What is the smallest value of $N$ that guarantees $\left|R_N\right|<0.01\text{?}$ Evaluate $S_N.$
2. The series
   
   $\begin{array}{cc}\hfill \pi & =6{\displaystyle \sum }_{n=0}^{\infty }\frac{\left(2n\right)\text{!}}{2^{4n+1}{\left(n\text{!}\right)}^2\left(2n+1\right)}\hfill \\ & =6\left(\frac{1}{2}+\frac{1}{2·3}{\left(\frac{1}{2}\right)}^3+\frac{1·3}{2·4·5}·{\left(\frac{1}{2}\right)}^5+\frac{1·3·5}{2·4·6·7}{\left(\frac{1}{2}\right)}^7+\text{⋯}\right)\hfill \end{array}$
   
   has been attributed to Newton in the late $1600\text{s}\text{.}$ The proof of this result uses the Maclaurin series for $f\left(x\right)={\text{sin}}^{\mathrm{-1}}x.$
   
   1. Prove that the series converges.
   2. Evaluate the partial sums $S_n$ for $n=5,10,20.$
   3. Compare $S_n$ to $\pi$ for $n=5,10,20$ and discuss the number of correct decimal places.
3. The series
   
   $\frac{1}{\pi }=\frac{\sqrt{8}}{9801}{\displaystyle \sum }_{n=0}^{\infty }\frac{\left(4n\right)\text{!}(1103+26390n)}{{\left(n\text{!}\right)}^4{396}^{4n}}$
   
   was discovered by Ramanujan in the early $1900\text{s}\text{.}$ William Gosper, Jr., used this series to calculate $\pi$ to an accuracy of more than $17$ million digits in the $\text{mid-}1980\text{s}\text{.}$ At the time, that was a world record. Since that time, this series and others by Ramanujan have led mathematicians to find many other series representations for $\pi$ and $1\text{/}\pi .$
   
   1. Prove that this series converges.
   2. Evaluate the first term in this series. Compare this number with the value of $\pi$ from a calculating utility. To how many decimal places do these two numbers agree? What if we add the first two terms in the series?
   3. Investigate the life of Srinivasa Ramanujan $(1887\text{–}1920)$ and write a brief summary. Ramanujan is one of the most fascinating stories in the history of mathematics. He was basically self-taught, with no formal training in mathematics, yet he contributed in highly original ways to many advanced areas of mathematics.