Pi is Irrational/Proof 2/Lemma
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Pi is Irrational: Lemma
Let $n \in \Z_{> 0}$ be a positive integer.
Let it be supposed that $\pi$ is irrational, so that:
- $\pi = \dfrac p q$
where $p$ and $q$ are integers and $q \ne 0$.
Let $A_n$ be defined as:
- $\ds A_n = \frac {q^n} {n!} \int_0^\pi \paren {x \paren {\pi - x} }^n \sin x \rd x$
Then:
- $A_n = \paren {4 n - 2} q A_{n - 1} - p^2 A_{n - 2}$
is a reduction formula for $A_n$.
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds \paren {x \paren {\pi - x} }^n\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x}\) | Power Rule for Derivatives and Chain Rule for Derivatives |
and let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \sin x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds -\cos x\) | Primitive of Sine Function |
Then:
\(\ds A_n\) | \(=\) | \(\ds \frac {q^n} {n!} \int_0^\pi \paren {x \paren {\pi - x} }^n \sin x \rd x\) | by definition | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {q^n} {n!} \paren {\bigintlimits {\paren {x \paren {\pi - x} }^n \paren {-\cos x} } 0 \pi - \int_0^\pi \paren {-\cos x} n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x} \rd x }\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {q^n} {n!} \int_0^\pi n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x} \cos x \rd x\) | as $\bigintlimits {\paren {x \paren {\pi - x} }^n \paren {-\cos x} } 0 \pi$ trivially evaluates to $0$ |
$\Box$
Let:
\(\ds u\) | \(=\) | \(\ds n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds n \paren {n - 1} \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi - 2 x}^2 + \paren {-2} n \paren {x \paren {\pi - x} }^{n - 1}\) | Power Rule for Derivatives, Product Rule for Derivatives and Chain Rule for Derivatives |
Now let:
\(\ds \frac {\d v} {\d x}\) | \(=\) | \(\ds \cos x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \sin x\) | Primitive of Cosine Function |
We note in passing that:
- $(1): \quad \paren {q \pi}^2 = q^2 \paren {\dfrac {p^2} {q^2} } = p^2$
which, by hypothesis, is an integer.
Then:
\(\ds A_n\) | \(=\) | \(\ds \frac {q^n} {n!} \int_0^\pi n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x} \cos x \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {q^n} {n!} \paren {\bigintlimits {n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x} \paren {\sin x} } 0 \pi - \int_0^\pi \paren {\sin x} \paren {n \paren {n - 1} \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi - 2 x}^2 - 2 n \paren {x \paren {\pi - x} }^{n - 1} } \rd x}\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {q^n} {n!} \paren {-\int_0^\pi \paren {\sin x} \paren {n \paren {n - 1} \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi - 2 x}^2 - 2 n \paren {x \paren {\pi - x} }^{n - 1} } \rd x}\) | as $\bigintlimits {n \paren {x \paren {\pi - x} }^{n - 1} \paren {\pi - 2 x} \paren {\sin x} } 0 \pi$ trivially evaluates to $0$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {q^n} {n!} \int_0^\pi \paren {2 n \paren {x \paren {\pi - x} }^{n - 1} - n \paren {n - 1} \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi - 2 x}^2 } \sin x \rd x\) | Sine of Integer Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {q^n} {n!} 2 n \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x - \frac {q^n} {n!} n \paren {n - 1} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi - 2 x}^2 \sin x \rd x\) | Linear Combination of Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 q \frac {q^{n - 1} } {\paren {n - 1}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x - q^2 \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi - 2 x}^2 \sin x \rd x\) | Definition of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 q A_{n - 1} - q^2 \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 2} \paren {\pi^2 - 4 x \pi + 4 x^2} \sin x \rd x\) | recalling $\ds A_{n - 1} = \frac {q^{n - 1} } {\paren {n - 1}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x$ and Square of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 q A_{n - 1} - \pi^2 q^2 \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 2} \sin x \rd x + 4 q^2 \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x\) | Linear Combination of Integrals | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 q A_{n - 1} - p^2 \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 2} \sin x \rd x + 4 \frac {\paren {n - 1} } {\paren {n - 1} } q^2 \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x\) | multiplying top and bottom by $\paren {n - 1}$, and $\pi^2 q^2 = p^2$ from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 q A_{n - 1} - p^2 A_{n - 2} + \paren {4 n - 4} q \frac {q^{n - 1} } {\paren {n - 1}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x\) | recalling $\ds A_{n - 2} = \frac {q^{n - 2} } {\paren {n - 2}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 2} \sin x \rd x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 q A_{n - 1} - p^2 A_{n - 2} + \paren {4 n - 4} q A_{n - 1}\) | recalling $\ds A_{n - 1} = \frac {q^{n - 1} } {\paren {n - 1}!} \int_0^\pi \paren {x \paren {\pi - x} }^{n - 1} \sin x \rd x$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {4 n - 2} q A_{n - 1} - p^2 A_{n - 2}\) |
$\blacksquare$