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When |r| < 1 the terms shrink fast enough for the series to converge, and the sum is a/(1 − r). The partial sum a(1 − rⁿ)/(1 − r) approaches that limit geometrically.
Sum to infinity
S = a / (1 − r), valid for |r| < 1
Partial sum
Sₙ = a(1 − rⁿ) / (1 − r)
8/(1 − 0.5) = 16.
The terms do not shrink, so the partial sums grow without limit or oscillate; the series diverges and has no sum.