One Third as Quotient of Sequences of Odd Numbers
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Theorem
- $\dfrac 1 3 = \dfrac {1 + 3} {5 + 7} = \dfrac {1 + 3 + 5} {7 + 9 + 11} = \dfrac {1 + 3 + 5 + 7} {9 + 11 + 13 + 15} = \cdots$
That is:
- $\forall n \in \Z_{> 0}: \dfrac 1 3 = \dfrac {\ds \sum_{k \mathop = 1}^n \paren {2 k - 1} } {\ds \sum_{k \mathop = n + 1}^{2 n} \paren {2 k - 1} }$
Proof
\(\ds \sum_{k \mathop = n + 1}^{2 n} \paren {2 k - 1}\) | \(=\) | \(\ds \sum_{k \mathop = 1}^{2 n} \paren {2 k - 1} - \sum_{k \mathop = 1}^n \paren {2 k - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 n}^2 - n^2\) | Odd Number Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 n^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sum_{k \mathop = 1}^n \paren {2 k - 1}\) | Odd Number Theorem |
The result follows.
$\blacksquare$
Historical Note
The result:
- $\dfrac 1 3 = \dfrac {1 + 3} {5 + 7} = \dfrac {1 + 3 + 5} {7 + 9 + 11} = \dfrac {1 + 3 + 5 + 7} {9 + 11 + 13 + 15} = \cdots$
was discovered by Galileo Galilei and published by him in $1615$.
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 / 3$