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