Adding smaller and smaller amounts can still total something: Picture walking half the remaining distance to a wall, then half of what's left, then half again. Each step is tiny — and the steps never stop — yet you never pass the wall. The distances 1, ½, ¼, ⅛, … add up to a finite total (here, 2).
This works whenever the common ratio r satisfies −1 < r < 1 (we write |r| < 1). The terms shrink toward zero fast enough that the running total closes in on a fixed number.
That fixed number is the sum to infinity.
If instead |r| ≥ 1 the terms never shrink (they stay the same size or grow), the total runs away to infinity, and there is no sum to infinity.
IB-style question — find the sum to infinity
A geometric series has first term u₁ = 12 and common ratio r = 0.5.
Find the sum to infinity.
Step by step
- First check convergence: is |r| < 1? Here r = 0.5, so |0.5| < 1 — yes, a sum to infinity exists.
- Write the formula for the sum to infinity.
- Substitute u₁ = 12 and r = 0.5.
- Work it out.
Final answer
S∞ = 24. The terms 12, 6, 3, 1.5, … keep being added but the total never goes past 24.
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First gate: is |r| < 1?: Before you ever write S∞ = u₁/(1 − r), ask does the series even settle down?
−1 < r < 1 (the terms shrink) → there is a sum to infinity.
r ≥ 1 or r ≤ −1 (the terms stay big or grow) → there is no sum to infinity; saying "S∞ = …" is wrong.
Many exam marks are lost by plugging an out-of-range r (like r = 2) into the formula and reporting a number. If |r| ≥ 1, the correct answer is the sum to infinity does not exist.
You can also run the formula backwards: if you're told S∞ and u₁, rearrange to find r.
IB-style question — drug building up in the body
A patient takes a 50 mg dose of a drug each day. By the next dose, only 30% of the amount present remains in the body (the rest is cleared). Over a long course of treatment the amount in the body just after a dose approaches a steady level.
Find this long-run steady amount, and comment on what it means for the patient.
Step by step
- Just after the newest dose, the body holds the new 50 mg plus the leftovers of every earlier dose. Each earlier dose has been multiplied by 0.30 once per day, so the leftovers form a geometric series with first term 50 and ratio r = 0.30.
- Check convergence: |0.30| < 1, so a long-run (steady) total exists.
- Apply the sum to infinity.
- Evaluate.
Final answer
The amount levels off at about 71.4 mg. In context: even with daily dosing the drug does not build up without limit — it stabilises near 71 mg, which is what lets doctors choose a safe steady dose.
IB-style question — find r from the sum to infinity
A convergent geometric series has first term 24 and sum to infinity 36.
Find the common ratio r.
Step by step
- Start from the formula and substitute what you know.
- Multiply both sides by (1 − r).
- Solve for the bracket, then for r.
- Check it converges.
Final answer
r = 1/3 (and since |1/3| < 1 the series genuinely converges, so the answer is valid).