The Infinite Sum of Mersenne Numbers

The Erdős–Borwein constant is a mathematical curiosity formed by adding up the reciprocals of every Mersenne number. Despite the complexity of these specific integers, the final sum is an irrational number.

A Mersenne number is defined as two raised to the power of a natural number minus one. When you take the reciprocal of every one of these numbers and add them all together, you get this mysterious constant. Because the sum is irrational, its decimal expansion never repeats or ends. This constant serves as a beautiful example of how simple rules in number theory can create infinite, non-repeating values.