This paper will look at the series that converge to pi, and which ones slowly converge to pi and the ones that converge to pi fast. And after looking at these different calculations of pi, I will attempt to find a series that converges to pi even faster. What lead me to do this topic was my passion for math, and curiosity of learning about series and limits to infinity. Pi is something that everybody knows as 3.14, but they have no idea how it came to be approximated as 3.14. It is a complex topic but once you take the time to sit down and look at it, it can be very interesting and beautiful math. First to understand this we will look at the history of π and its calculation. After looking at the history I will then explore the different calculations for
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2. What are series? Series are just terms of a sequence; you are just adding them up. For example 1+2+3+4+5+6…. is a series. This is just an easy series, a much more complicated series is ∑_(n=0)^∞▒x^n/n! which is a called a Taylor series. There are much more complicated series and are very useful in math, for example these series may be used to calculate pi or e. First we will take a look at all the series that converge to pi and we will classify them as either converging to pi fast or slow and why.
2.1 Series converging to pi
Gregory’s formula
The first formula ill look at is Gregory’s formula for pi, which was obtained by plugging 1 for x. Gregory’s formula states: equation (9) tan^(-1)1= 1-1^3/3+1^5/5-1^7/7+⋯ tan^(-1)1 That is just π/4
Thus Gregory’s formula π/4 =1-1^3/3+1^5/5-1^7/7+⋯
Machin’s formula
The first formula ill look at is the Machin’s formula, by John Machin and he computed pi to 100 decimal
2. What are series? Series are just terms of a sequence; you are just adding them up. For example 1+2+3+4+5+6…. is a series. This is just an easy series, a much more complicated series is ∑_(n=0)^∞▒x^n/n! which is a called a Taylor series. There are much more complicated series and are very useful in math, for example these series may be used to calculate pi or e. First we will take a look at all the series that converge to pi and we will classify them as either converging to pi fast or slow and why.
2.1 Series converging to pi
Gregory’s formula
The first formula ill look at is Gregory’s formula for pi, which was obtained by plugging 1 for x. Gregory’s formula states: equation (9) tan^(-1)1= 1-1^3/3+1^5/5-1^7/7+⋯ tan^(-1)1 That is just π/4
Thus Gregory’s formula π/4 =1-1^3/3+1^5/5-1^7/7+⋯
Machin’s formula
The first formula ill look at is the Machin’s formula, by John Machin and he computed pi to 100 decimal