Leibniz formula for Pi. When it is solved for Pi one gets:
Pi=4/1-4/3+4/5-4/7...
And so on since it's an infinite series. Since one can see a pattern that can easily be defined I concluded that a simple program using a loop could be made. I decided on using JavaScript for its ease of use and use a for loop. The program I made at first was mainly:

var Pi=0;
var n=1;
for (i=0;i<=1000;i++)
{
Pi=Pi+(4/n)
n=n+2
Pi=Pi-(4/n)
n=n+2
}

Where the variable "Pi" is the value of Pi, "i" is the number of times the loop is repeated, and "n" is just a variable to help in the formula. This I however found could further simplified to:

var Pi=0;
var n=1;
for (i=0;i<=1000;i++)
{
Pi=Pi+(4/n)-(4/(n+2))
n=n+4
}

Of course the more times the loop is repeated, the more accurate your value of Pi gets, and while 1000 may seem like a large number, the value of Pi which it calculated is wrong after the 3rd decimal point. I've had it repeat over 100,000 times and it gets more accurate, however, remember that making the number of repeats can slow down and even crash your browse

The maximum decimal places in javascript is limited to 15. So you cannot get more than 15 decimal places.

Viete's Series
The first infinite sequence discovered in Europe was an infinite product, found by French mathematician François Viète in 1593:

Wallis's Series
The second infinite sequence, found in Europe by John Wallis in 1655, was also an infinite product:

Leibniz's Series
Madhava of Sangamagrama, a Indian mathematician, formulated a series that was rediscovered by scottish mathematician James Gregory in 1671, and by Leibniz in 1674:

π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 +4/13 -........

Nilakantha's Series
An infinite series for PI published by Nilakantha in the 15th century is: