Preview
Sum of Infinite Geometric Series
IB DP Mathematics: Applications and Interpretation | SL & HL | Number and Algebra | Topic 1.11
📋 Learning Intentions
- State the condition for an infinite geometric series to converge, using the common ratio \( r \).
- Apply the formula \( S_\infty = \dfrac{u_1}{1 - r} \) to calculate the sum of a convergent infinite geometric series.
- Determine whether a given geometric series converges or diverges by testing \( |r|
- Find unknown terms or the common ratio given the infinite sum and one other piece of information.
- Interpret the sum of an infinite geometric series in real-world contexts such as drug dosage, bouncing balls, and recurring decimals.
Real-World Introduction
A ball is dropped from a height of 2 metres. Each time it bounces, it rises to 60% of its previous height. The first bounce reaches 1.2 m, the second 0.72 m, the third 0.432 m, and so on. If you add up every bounce height — infinitely many of them — you get a finite total distance. This is not intuitive: how can infinitely many positive numbers add up to something finite?
The same mathematics governs drug accumulation in the bloodstream. A patient takes a 200 mg tablet every 12 hours, and the body eliminates 70% of the drug before the next dose. After many doses, the amount of drug in the body stabilises at a fixed level — the sum of an infinite geometric series. Pharmacologists use this calculation to set safe dosing intervals.
🌍 Why This Matters Infinite geometric series underpin financial perpetuities (bonds that pay forever), ecological models of population stabilisation, and signal processing in engineering. The convergence condition \( |r| Recall that a geometric sequence has a constant ratio between consecutive terms. When the terms of a geometric sequence are added together, the result is a geometric series. For a finite series with \( n \) terms, the sum always exists. The interesting question is: what happens when \( n \to \infty \)?
Consider the series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \). The partial sums are 1, 1.5, 1.75, 1.875, … — they are increasing but appear to be approaching 2. This behaviour — where partial sums