Series Converging to Pi

Series Converging to Pi

lundy 18 Nov 2008 06:33

Chip and Josh in class today gave the series $\sum_{i=1}^{k} (-1)^{i-1}\frac{4} {2 i - 1} = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} ...$ as an example of a series that converges to $\pi$.

After doing some further research, I stumbled upon two other series that have convergence related to $\pi$:

Ramanujan's Formula

(1)

\begin{align} \pi = 2 \sqrt{3} \sum_{n=0}^{\infty}{\frac{(-1)^{n}}{(2n+1)3^{n}}} \end{align}

Chudnovsky Algorithm

(2)

\begin{align} \frac{1}{\pi} = 12\sum_{k=0}^{\infty}{\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+\frac{3}{2}}} \end{align}

The Chudnovsky Algorithm is based on a rapidly converging hypergeometric series. It was used to generate over a billion digits of $\pi$! Mathematica uses it today to calculate $\pi$.

Re: Series Converging to Pi

jonas2 18 Nov 2008 23:29

Similarly, I have found other series converging to various multiples of $\pi$ in one of my other classes. Sadly, this of course requires methods outside the scope of this class but it's still pretty interesting that a large sum of rationals add to such a peculiar irrational.

$\frac{\pi^2}{8} = \[\sum_{n=0}^{\infty} \frac{1}{(2n+1)^2} \]$

This sum can be derived with the Fourier sine series of $f(x)=1$ on $(0,\pi)$.

$\frac{\pi^2}{6} = \[\sum_{n=1}^{\infty} \frac{1}{n^2} \]$

This sum can be derived with the Fourier sine series of $f(x)=x$ on $(0,l)$.

$\frac{\pi^4}{90} = \[\sum_{n=1}^{\infty} \frac{1}{n^4} \]$

This sum can be derived with the Fourier cosine series of $f(x)=x^2$ on $(0,l)$.

(Additionally, this required the use of Parseval's Equality).