Infinite Exponential Series for Fractions

Fractions can be represented by an infinite series. The rate at which this series converges is log(y + 1) decimal digits per term.

Equation

 * x/y = x/(y + 1) + x/(y + 1)2 + x/(y + 1)3 + ...

Manual Calculation

 * (Ex. x = 2; y = 7)
 * 2/8 = 0.25
 * 2/82 = 0.03125
 * 2/83 = 0.00390625
 * 2/84 = 0.00048828125
 * 2/85 = 0.00006103515
 * 2/8 + 2/82 + 2/83 + 2/84 + 2/85 = 0.25 + 0.03125 + 0.00390625 + 0.00048828125 + 0.00006103515 = 0.2857055664
 * 2/7 = 0.2857142857...
 * (Ex. x = 12; y = 5)
 * 12/6 = 2
 * 12/62 = 0.33333333333
 * 12/63 = 0.05555555555
 * 12/64 = 0.00925925925
 * 12/65 = 0.00154320987
 * 12/6 + 12/62 + 12/63 + 12/64 + 12/65 = 2 + 0.33333333333 + 0.05555555555 + 0.00925925925 + 0.00154320987 = 2.399691358
 * 2/7 = 2.4

Alternate Form

 * x/y = x/(y - 1) - x/(y - 1)2 + x/(y - 1)3 - ...

This is an alternating series that converges similarly to the above. It alternates for positive numbers and converges geometrically for negative numbers.
 * (Ex. x = 23; y = 11)
 * 23/10 = 2.3
 * 23/102 = 0.23
 * 23/103 = 0.023
 * 23/104 = 0.0023
 * 23/105 = 0.00023
 * 23/10 - 23/102 + 23/103 - 23/104 + 23/105 = 2.3 - 0.23 + 0.023 - 0.0023 + 0.00023 = 2.09093
 * 23/11 = 2.0909090909...