Pi as Sum of Alternating Sequence of Products of 3 Consecutive Reciprocals

Theorem

 * $\dfrac {\pi - 3} 4 = \dfrac 1 {2 \times 3 \times 4} - \dfrac 1 {4 \times 5 \times 6} + \dfrac 1 {6 \times 7 \times 8} \cdots$

Proof
Let $f: \R \to \R$ be the real function defined as:


 * $\forall x \in \R: \map f x = x^1 - x^3 + x^5 - x^7 + x^9 - x^{11} + x^{13} - x^{15} \cdots$

We have:

Integrating 3 times will give us the desired series.

Lemma
Substitute $x = 1$: