Leibniz's Formula for Pi/Proof by Taylor Expansion

Theorem

 * $\displaystyle \frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \frac 1 9 - \cdots $

That is:
 * $\displaystyle \pi = 4 \sum_{k \ge 0} \left({-1}\right)^k \frac 1 {2 k + 1}$

Proof
From Taylor Expansion of Arctangent Function, we obtain:


 * $\displaystyle \arctan x = x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \frac {x^9} 9 - \cdots$

Substituting $x = 1$ gives the required result.