1 + 2 + 4 + 8 + ⋯

The series \(1 + 2 + 4 + 8 + \ldots\) is a geometric series where the first term \(a = 1\) and the common ratio \(r = 2\). The sum of an infinite geometric series can be calculated using the formula: \[ S = \frac{a}{1 - r} \] This formula is valid only for \(|r| < 1\).