Daleth function (Martinez)

The daleth function, as described by Gilbert Martinez on 27 Sep 2018, allows one to compute the sum of any geometric series.

$\daleth (a, r) = a \sum_{n=0}^{\infty} r^{n} = \frac{at}{t-s} \iff |r| < 1 \and r = \frac{s}{t}$

Here, a is the first term of the series, and r is the fraction that is exponentially multiplied to each term.

$a \sum_{n=0}^{\infty} r^{n} = a + ar + ar^{2} + ar^{3} + ...$

Here are a few examples of the daleth function in action.

$\daleth (1,{\frac {1}{2}})=\sum _{n=0}^{\infty }({\frac {1}{2}})^{n}={\frac {1*2}{2-1}}={\frac {2}{1}}=2$

$\daleth ({\frac {5}{2}},{\frac {3}{8}})={\frac {5}{2}}\sum _{n=0}^{\infty }({\frac {3}{8}})^{n}={\frac {{\frac {5}{2}}*8}{8-3}}={\frac {\frac {40}{2}}{5}}={\frac {20}{5}}=4$

$\daleth ({\frac {10}{3}},{\frac {4}{9}})={\frac {10}{3}}\sum _{n=0}^{\infty }({\frac {4}{9}})^{n}={\frac {{\frac {10}{3}}*9}{9-4}}={\frac {\frac {90}{3}}{5}}={\frac {30}{5}}=6$

$\daleth ({\frac {1}{2}},{\frac {5}{7}})={\frac {1}{2}}\sum _{n=0}^{\infty }({\frac {5}{7}})^{n}={\frac {{\frac {1}{2}}*7}{7-5}}={\frac {\frac {7}{2}}{2}}={\frac {7}{4}}$

The daleth hypothesis, proposed the same day, suggests that every rational number can be computed with the daleth function, although this hasn't been proven yet.

Martinez proposed that the proof would consist of two parts:

1. Proving that the daleth function can generate any whole number.
2. Proving that the daleth function can generate any fraction.

Observations

Martinez noted that, using only rational numbers, the daleth function could never produce an irrational number, since the numbers used would always be whole number ratios.

He also noted that any whole number can be trivially generated using the following:

$\daleth (1, \frac{x-1}{x}) = \frac{1 * x}{x - (x - 1)} = \frac{x}{x - x + 1} = \frac{x}{1} = x$

For his proof, however, he wants to find nontrivial ways to generate any whole number.