Convergence

Definition

An infinite geometric series converges when the common ratio satisfies −1 < r < 1. In that case the sum to infinity is S∞ = a/(1 − r). If |r| ≥ 1 the series diverges and has no finite sum.

Worked example

For 8 + 4 + 2 + … , r = ½, so S∞ = 8/(1 − ½) = 16.

Where it appears in the exam

Paper 1 asks for the values of x that make a geometric series converge, or for the sum to infinity of a convergent series.

Step-by-step method

1. Find r = T₂ ÷ T₁.
2. Set up the condition −1 < r < 1.
3. Solve the inequality if r contains x.
4. If the series converges, calculate S∞ = a/(1 − r).

Common mistake

Including r = 1 or r = −1 in the condition: the inequality is strict.

Related sequences terms

• Arithmetic sequence
• Geometric sequence
• Sigma notation

Frequently asked questions

What does convergence mean in Grade 12 Maths?

An infinite geometric series converges when the common ratio satisfies −1 < r < 1. In that case the sum to infinity is S∞ = a/(1 − r). If |r| ≥ 1 the series diverges and has no finite sum.

Which NSC paper tests convergence?

Convergence belongs to the Sequences section of CAPS Grade 12 Mathematics, which is examined in Paper 1.

How do I practise questions involving convergence?

Snap a question that uses convergence into the Snap&Learn AI solver, or work through the sequences topic guide and NSC past papers.