# How to, using Taylor series approximation , estimate the value of π, when arctan(x) ≈ x-x3/3+x5/5-x7/7 ?

##### 2 Answers

let

Using the specified TS approximation we get

#### Explanation:

Using the given Taylor Series approximation (truncation) we have:

# arctanx ~~ x - x^3/3 + x^5/5 - x^7/7 #

Using the well known result:

# tan (pi/4) =1 => arctan1=pi/4 #

So, substituting

# pi/4 ~~ 1 - 1/3 + 1/5 - 1/7 #

# \ \ \ \ = (3*5*7 - 5*7 + 3*7 - 3.5)/(3*5*7) #

# \ \ \ \ = (105 - 35 + 21-15)/(105) #

# \ \ \ \ = (76)/(105) #

Thus:

# pi ~~ 4 * (76)/(105) #

# \ \ \ \ = 304/105 #

# \ \ \ \ ~~= 2.90 \ \ \ # (3sf)

A (not so) interesting fact is that using this particular Taylor Series and method to approximate