Algebra 2 · Unit 9 · Topic 9.4
Infinite Geometric Series and Convergence
An infinite geometric series has finite sum a₁/(1 − r) when |r| < 1, because the contribution of later terms tends to zero.
Definition and explanation
An infinite geometric series has finite sum a₁/(1 − r) when |r| < 1, because the contribution of later terms tends to zero. For a nonzero first term, it does not converge when |r| ≥ 1.
Worked example
The series 9 + 3 + 1 + 1/3 + … has ratio 1/3. Its sum is 9/(1 − 1/3) = 27/2 = 13.5. The first three terms total 13, leaving a remaining sum of 0.5.
Common mistake
The infinite-sum formula must not be applied to a ratio with absolute value at least 1.
Key takeaway
An infinite geometric series has finite sum a₁/(1 − r) when |r| < 1, because the contribution of later terms tends to zero.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.