Find the sum of all the terms in the series 1, 1/2, 1/4, ...... (an infinite…

Concept: An infinite geometric series with first term a and common ratio r, where |r| < 1, converges to a finite sum given by S = a / (1 - r). This follows from the partial-sum formula Sn = a(1 - rn) / (1 - r): as n grows without bound and |r| < 1, rn approaches 0, so the partial sums approach a / (1 - r).

1. Identify the first term and common ratio from the given series 1, 1/2, 1/4, ......: a = 1, and r = 1/2 (each term is half the one before it).

2. Confirm convergence: since |r| = 1/2 < 1, the series has a finite sum, so the formula S = a / (1 - r) applies.

3. Substitute the values into the formula: S = 1 / (1 - 1/2) = 1 / (1/2) = 2.

Cross-check: the running partial sums — 1, 1.5, 1.75, 1.875, 1.9375, ...... — steadily approach 2 without ever reaching or exceeding it, confirming the formula's result.