Approximating PI: Improving Arctan (1/√3) Approximation

[linyen416]

by writing arctan as a power series, you can obtain an approximation to arctan (1/root 3) and hence approximate pi. I've done this by multiplying my answer for arctan (1/root3) by 6 to get 3.14... radians.

I would like to know How can this approximation be improved?


thanks!

 

[schroder]

by writing arctan as a power series, you can obtain an approximation to arctan (1/root 3) and hence approximate pi. I've done this by multiplying my answer for arctan (1/root3) by 6 to get 3.14... radians.

I would like to know How can this approximation be improved?


thanks!

This is an interesting question because in order to calculate the approximate value of pi by using an arctan series, you first need to calculate the value of the arctan by using an infinite series.

The power series for arctan (x) is:

x – x^3/3 + x^5/5 – x^7/7 . . .

The only problem I see with your method is your choice of x as being 1 / sqrt 3. This is itself an irrational number so you would first need to either know its value to many decimal places, or calculate that from another infinite series such as the binomial expansion, which greatly complicates your work.

I would recommend you follow the example of William Shanks who spent some twenty years on this problem from 1853 to 1873! He used an equation known as Machin’s formula which is based on the value of two arctangents;

pi/4 = 4 arctan (1/5) – arctan (1/239).

This has the advantage that the values of x are at least rational numbers and they are both small numbers so the series converges quickly. A good exercise for a rainy day!