Find the sum of all terms in the series 1, 1/2, 1/4 ..... ?

Concept: For an infinite geometric series with first term a and common ratio r where |r| < 1, the sum of all terms (sum to infinity) is given by the formula S∞ = a / (1 − r). This follows from taking the limit of the partial-sum formula Sn = a(1 − rn)/(1 − r) as n → ∞, since rn → 0 when |r| < 1.

Application:

1. Identify the first term and common ratio of the series 1, 1/2, 1/4, .....: a = 1, and r = (1/2) ÷ 1 = 1/2.

2. Since |r| = 1/2 < 1, the series converges, so the sum-to-infinity formula applies.

3. Substitute a = 1 and r = 1/2 into S∞ = a / (1 − r): S∞ = 1 / (1 − 1/2) = 1 / (1/2).

4. Simplify: 1 / (1/2) = 2.

Cross-check: Adding the first several partial sums confirms convergence toward this value: 1 + 1/2 + 1/4 + 1/8 + 1/16 = 1.9375, and the partial sums keep approaching 2 as more terms are added, matching the formula's result.

Answer: Hence, the sum of all terms in the series is 2.