Leibniz's Formula for Pi/Elementary Proof

Theorem

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

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

Proof
First we note that:


 * $\displaystyle (1): \quad \frac 1 {1+t^2} = 1 - t^2 + t^4 - t^6 + \cdots + t^{4n} - \frac {t^{4n + 2}}{1+t^2}$

which is demonstrated here.

Now consider the real number $x \in \R: 0 \le x \le 1$.

We can integrate expression $(1)$ WRT $t$ from $0$ to $x$:


 * $\displaystyle (2) \qquad \int_{0}^{x}\frac {\mathrm{d}{t}}{1+t^2} = x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots + \frac {x^{4n + 1}}{4n + 1} - R_n \left({x}\right)$

where:
 * $\displaystyle R_n \left({x}\right) = \int_{0}^{x} \frac {t^{4n + 2}}{1+t^2}\mathrm{d}{t}$

From Even Powers are Positive we have that $t^2 \ge 0$ and so $1 \le 1 + t^2$.

From Relative Sizes of Definite Integrals, we have:
 * $\displaystyle 0 \le R_n \left({x}\right) \le \int_{0}^{x} t^{4n + 2} \mathrm{d}{t}$

that is:
 * $\displaystyle 0 \le R_n \left({x}\right) \le \frac {x^{4n + 3}}{4n + 3}$

But as $0 \le x \le 1$ it is clear that $\dfrac {x^{4n + 3}}{4n + 3} \le \dfrac 1 {4n + 3}$.

So:
 * $0 \le R_n \left({x}\right) \le \dfrac 1 {4n + 3}$

From Basic Null Sequences and the Squeeze Theorem, $\dfrac 1 {4n + 3} \to 0$ as $n \to \infty$.

This leads us directly to:


 * $\displaystyle (2): \quad \int_{0}^{x}\frac {\mathrm{d}{t}}{1+t^2} = x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \frac {x^9} 9 \cdots$

But from Derivative of Arctangent Function, we also have that:
 * $\displaystyle \frac{\mathrm{d}{}}{\mathrm{d}{x}} \arctan t = \frac 1 {1+t^2}$

and thence from the Fundamental Theorem of Calculus we have:
 * $\displaystyle \arctan x = \int_{0}^{x}\frac {\mathrm{d}{t}}{1+t^2}$

From $(2)$ it follows immediately that:
 * $\displaystyle (3): \quad \arctan x = x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \frac {x^9} 9 \cdots$

Now all we need to do is plug $x = 1$ into $(3)$.

Comment
Note that we did not just take the Sum of Infinite Geometric Progression:
 * $\displaystyle \frac 1 {1 - \left({-t^2}\right)} = 1 + \left({-t^2}\right) + \left({-t^2}\right)^2 + \left({-t^2}\right)^3 + \cdots$

and integrate it term by term, as we have not at this stage proved that this is permissible.