A geometric series has a sum to infinity only when |r| < 1, and then the sum is S∞ = a/(1 − r). The order matters: check the condition first, then use the formula.
This lesson follows recovering a common ratio and belongs to arithmetic and geometric series.
Why does the ratio decide everything?
Each term is r times the one before. If r lies between −1 and 1, every term is smaller in size than the last, so the additions become tiny and the running total settles.
If r is 1, every term is the same and the total grows without limit. If |r| is greater than 1, the terms themselves grow. In both cases there is no finite sum to infinity.
The finite sum of a geometric progression is Sn = a(1 − rn)/(1 − r), also written a(rn − 1)/(r − 1). When |r| < 1, rn tends to zero, and Sn tends to a/(1 − r).
How do you use it, step by step?
- Find r by dividing a term by the one before.
- Test the condition: is |r| < 1? If not, write that no sum to infinity exists.
- Substitute into a/(1 − r), taking care with brackets when r is negative or a fraction.
- Sense-check: the answer should look like a plausible total of the first few terms.
Worked example
Find the sum to infinity of 18 + 12 + 8 + …
Step 1, ratio: r = 12 ÷ 18 = 2/3. Check the next pair: 8 ÷ 12 = 2/3.
Step 2, condition: |2/3| < 1, so the sum to infinity exists.
Step 3, formula: S∞ = 18 ÷ (1 − 2/3) = 18 ÷ (1/3) = 18 × 3 = 54.
Step 4, sense-check: the first five terms are 18, 12, 8, 5.33, 3.56 and they add to about 46.9. The remaining terms are shrinking, so a limit of 54 is plausible.
A second case, with a negative ratio: for 8 − 4 + 2 − 1 + …, r = −1/2 and S∞ = 8 ÷ (1 + 1/2) = 8 ÷ 3/2 = 16/3. Note the bracket: 1 − (−1/2) = 3/2, not 1/2.
The mistake to watch for
The slip is to use the formula without testing r.
Mistaken working: for 2 + 6 + 18 + …, r = 3, so S∞ = 2 ÷ (1 − 3) = 2 ÷ (−2) = −1.
A series of positive terms can never add to a negative number. The formula has produced nonsense because the condition was never checked.
The correction is to stop at step 2. Here |r| = 3, which is greater than 1, so the series has no sum to infinity. Whenever an infinite sum looks odd, such as negative for a positive series, the first thing to check is |r|.
Check yourself
Try these on paper, then open each answer.
1. Find the sum to infinity of 40 + 10 + 2.5 + …
Show answer
r = 10 ÷ 40 = 1/4, and |1/4| < 1. S∞ = 40 ÷ (1 − 1/4) = 40 ÷ 3/4 = 160/3, which is 53 1/3 (about 53.3).
Check: the first four terms add to 40 + 10 + 2.5 + 0.625 = 53.125, already close.
2. A geometric series has first term 12 and sum to infinity 16. Find r.
Show answer
12 ÷ (1 − r) = 16, so 1 − r = 12/16 = 3/4. Hence r = 1/4, which satisfies |r| < 1.
Check: 12 ÷ (3/4) = 16.
3. A geometric series has first term 7 and common ratio 2x − 1. Find the range of x for which the series has a sum to infinity.
Show answer
We need |2x − 1| < 1, so −1 < 2x − 1 < 1. Adding 1 to each part gives 0 < 2x < 2, so 0 < x < 1.
Check: x = 0.5 gives r = 0, and x = 1 gives r = 1, which is not allowed, so the boundaries are excluded.
Where this leads next
Once the condition is automatic, try mixing finite terms and sums in a combined term-and-sum problem. The sequence and series laboratory flags when |r| is too large for an infinite sum, which makes a useful check on your own examples.
If a question like the third one makes you hesitate over inequality signs, that is worth a short session in online one-to-one Additional Mathematics tuition.