Leibniz's Formula for Pi/Leibniz's 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

 * LeibnizFormula.png

The area $OAT$ is a quarter-circle whose area is $\dfrac \pi 4$ by Area of Circle.

Now consider the area $C$ of the segment $OPQT$, bounded by the arc $OT$ and the chord $OT$.

Consider the area $OPQ$, bounded by the line segments $OP$ and $OQ$ and the arc $PQ$.

As $P$ and $Q$ approach each other, the arc $PQ$ tends towards the straight line segment $\mathrm d s = PQ$.

We can therefore consider the area $OPQ$ as a triangle.

We extend the line segment $PQ$ and drop a perpendicular $OR$ to $O$.

Using Area of Triangle in Terms of Side and Altitude, we see that the area $\mathrm d C$ of $\triangle OPQ$ is given by:
 * $\displaystyle \mathrm d C = \triangle OPQ = \frac {OR \cdot PQ} 2 = \frac {OR \cdot \mathrm d s} 2$

We also note from elementary Euclidean geometry that $\triangle ORS$ is similar to the small triangle on $PQ$.

Thus:
 * $\displaystyle \frac {\mathrm d s}{\mathrm d x} = \frac {OS} {OR} \iff OR \cdot \mathrm d s = OS \cdot \mathrm d x$

Thus:
 * $\displaystyle \mathrm d C = \frac {OS \cdot \mathrm d x} 2 = \frac {y \mathrm d x} 2$

where $y = OS$.

We set the horizontal coordinate of $P$ as equal to $x$.

Thus the total area $C$ is equal to the total of all the areas of these small triangles as $x$ increases from $0$ to $1$.

So:
 * $\displaystyle C = \int \mathrm d C = \frac 1 2 \int_0^1 y \mathrm d x$

Now we use Integration by Parts to swap $x$ and $y$:
 * $\displaystyle C = \left[{\frac 1 2 x y}\right]_0^1 - \frac 1 2 \int_0^1 x \mathrm d y = \frac 1 2 - \frac 1 2 \int_0^1 x \mathrm d y$

It can be seen that the limits on this new integral have to be $0$ and $1$ from the geometry of the situation.

Now we note that:
 * $y = \tan \dfrac \phi 2$
 * $x = 1 - \cos \phi = 2 \sin^2 \dfrac \phi 2$ (from Double Angle Formula for Cosine: Corollary 2)

Thus:

and so:
 * $\displaystyle \frac x 2 = \frac {y^2}{1 + y^2}$

Using Sum of Geometric Progression:
 * $\displaystyle \frac {y^2}{1 + y^2} = y^2 - y^4 + y^6 - y^8 + \cdots$

This gives us:
 * $\displaystyle C = \frac 1 2 - \int_0^1 \left({y^2 - y^4 + y^6 - y^8 + \cdots}\right) \mathrm d y$

Remember, $C$ is the area of the segment $OPQT$.

Now we add to it the area of $\triangle OTA$, which trivially equals $\dfrac 1 2$, to get the area of the quarter circle which we know as equal to $\dfrac \pi 4$.

Putting it all together, this gives us:
 * $\displaystyle \frac \pi 4 = 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \frac 1 9 - \cdots$