Geometric series: when does the sum-to-infinity actually exist?

Published 2026-04-08 · Written by Pete Bromfield

Every A-Level student can recite $S_\infty = \dfrac{a}{1 - r}$. Fewer can explain when it exists. That "when" is a reasoning mark.

The condition

$S_\infty$ exists iff $|r| < 1$. If $|r| \geq 1$, the terms don't shrink, and the sum diverges to infinity (or oscillates).

Worked example — with the reasoning mark called out

The sum to infinity of a geometric series is 8. The first term is 3. Find the common ratio, stating the condition on $r$.

1. $\dfrac{3}{1 - r} = 8 \Rightarrow 3 = 8 - 8r \Rightarrow r = \tfrac{5}{8}$.
2. Since $|r| = \tfrac{5}{8} < 1$, the sum to infinity exists.

That second line is the reasoning mark. Miss it and you lose one out of three. Most students just quote the value of $r$ and move on.

Drill

10 GP sum-to-infinity drills on Skills Practice → Sequences and Series.