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Ever since the discovery of Pi, Mathematicians have been obsessed with finding methods of approximating Pi. I think I've a unique way of doing so via the Newton-Raphson.

Newton-Raphson Formula:

Let ## ƒ(x)=Sin(x) ⇒ ƒ'(x)=Cos(x) ⇒ X_n= X_{n-1} - tan(X_{n-1})##

For example: Let ##X_0=X ⇒ X_3= (X-tan(X)-tan(X-tan(X))) - tan(X-tan(X)-tan(x-tan(X)))##

This cannot be calculated without a calculator therefore we can introduce a taylor expansion:

##Tan(x)≈x+x^3/3+(2 x^5)/15+(17 x^7)/315+...##

I'd appreciate any feedback on this method and ways to simplify/generalise my equation.

Newton-Raphson Formula:

Let ## ƒ(x)=Sin(x) ⇒ ƒ'(x)=Cos(x) ⇒ X_n= X_{n-1} - tan(X_{n-1})##

For example: Let ##X_0=X ⇒ X_3= (X-tan(X)-tan(X-tan(X))) - tan(X-tan(X)-tan(x-tan(X)))##

This cannot be calculated without a calculator therefore we can introduce a taylor expansion:

##Tan(x)≈x+x^3/3+(2 x^5)/15+(17 x^7)/315+...##

**NOTE**: For this to work the value of ##X_0 ## must be close to Pi, the rate of convergence is proportional to how close your value of ##X_0## is to Pi.I'd appreciate any feedback on this method and ways to simplify/generalise my equation.

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