value of π by Gregory-Leibniz series in fortran

An infinite sum series first given by Madhava and later rediscovered by Gregory and Leibniz is expressed as, 

`\sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = \frac{\pi}{4}`
or, `\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}`
`\therefore \pi = 4 \times \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}`

Where n  = 0,1,2,3,4,5, ..................................,  positive integer.

For n =0

`\pi = 4 \times 1`

For n = 0,1

   `\pi = 4 \times (1- \frac{1}{3})`

For n = 0,1,2

`\pi = 4 \times (1 - \frac{1}{3} + \frac{1}{5})`

For n = 0,1,2,3

`\pi = 4 \times (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7})` and so on.

Continuiing this process  up to `n \approx \infty`,  you obviously get,

   `\pi =4 \times ( 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + ....................)     (2)`

This is called Gregory-Leibniz series to estimate value of `\pi`. 

fig. 1. n vs. `\pi` plot

n gives number of terms in the series. 

To get fortran code for estimation of value of `\pi` by using Gregory-Liebniz series, click on the given button :  Get Files

Watch Video

Learn to create gif animation in gnuplot software: Click Here.