Calculating Pi

Euler’s Identity for pi

Euler noticed that the sum of the reciprocols of the squared intergers are: \[ \frac{\pi^2}{6} = \sum_{n=1}^{\infty} \frac{1}{n^2} \\ \pi = \sqrt{6\sum_{n=1}^{\infty} \frac{1}{n^2}} \]

Let’s see if this is true.

options(digits = 22)
pi #r's default value for pi.

## [1] 3.1415926535897931

n = c(1:10)
PI = (sum(6/n^2))^0.5
PI

## [1] 3.0493616359820699

pi-PI

## [1] 0.092231017607723231

n = c(1:100)
PI = (sum(6/n^2))^0.5
PI

## [1] 3.1320765318091062

pi-PI

## [1] 0.009516121780686948

n = c(1:10000)
PI = (sum(6/n^2))^0.5
PI

## [1] 3.1414971639472093

pi-PI

## [1] 9.5489642583768131e-05

n = c(1:100000000)
PI = (sum(6/n^2))^0.5
PI

## [1] 3.1415926440404967

pi-PI

## [1] 9.5492964646837208e-09