1.11 Infinite geometric series

Learning objectives

• Explain what it means for an infinite series to converge or diverge, using partial sums.

• Show that the partial sums of a geometric series approach $\frac{u_1}{1 - r}$ when $|r| < 1$.

• Calculate the sum to infinity of a geometric series, exactly where possible.

• Determine whether the sum to infinity of a geometric series exists, and justify the decision with the common ratio.

• Find the values of a parameter for which the sum to infinity exists, by solving $|r| < 1$.

• Find the first term or the common ratio from a given sum to infinity, rejecting any ratio with $|r| \ge 1$.

• Find the least number of terms for which a partial sum is within a given amount of the sum to infinity, using technology.

• Calculate the total distance travelled by a bouncing ball and comment on the validity of the model.

• Interpret sums to infinity in contexts such as repeated spending, drug doses and fractals.

Introduction

A geometric series can be added term by term for as long as one likes: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$

Infinite sums describe processes that repeat forever with each round smaller than the last: a ball that bounces lower and lower, money that is re-spent through an economy, a medicine that is topped up each day while the body removes part of it, a shape that gains ever smaller triangles. In each case the sum to infinity gives the long-term total or ceiling, which is often the most useful single number in the model.

Adding infinitely many terms

Zeno's dichotomy

Zeno's dichotomy paradox, from ancient Greece in the 5th century BCE, claims that a runner can never finish a race. Before reaching the finish, the runner must cover half the distance; then half of what remains; then half of what remains after that, and so on without end. Infinitely many tasks, Zeno claimed, cannot be completed. In modern terms the runner covers

of the course, a geometric series with first term $\frac{1}{2}$ and common ratio $\frac{1}{2}$. The halving of a unit square shows the same sum as areas: each new rectangle takes half of the area still empty, and the rectangles fill the square ever more completely.

After $n$ pieces the covered area is $1 - \left(\frac{1}{2}\right)^{n}$, because the uncovered part halves each time. The uncovered area becomes as small as one likes, smaller than any positive number once $n$

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Infinite series and partial sums

An infinite series is written with three dots, or with sigma notation and $\infty$ as the upper limit: $\sum_{k=1}^{\infty} u_k = u_1 + u_2 + u_3 + \cdots$

No one can add infinitely many numbers one at a time, so the value of an infinite series is defined through its partial sums $S_1, S_2, S_3, \ldots$, where $S_n$

The two rows behave very differently. The first approaches 1 more closely with every term; the second grows without limit, so no number can be called its sum.

When the partial sums approach no single number, the series has no sum. This happens when they grow without bound, as in $1 + 2 + 4 + \cdots$, and also when they keep jumping between values, as in $1 - 1 + 1 - 1 + \cdots$

The sum to infinity of a geometric series

Deriving the formula

The sum of the first $n$ terms of a geometric series, from subtopic 1.3, is

Only one part of this formula depends on $n$: the power $r^{n}$. What happens to $S_n$ as $n$ grows therefore depends entirely on what happens to $r^{n}$

When $-1 < r < 1$, repeated multiplication by a number smaller than 1 in size makes $r^{n}$ shrink towards 0, whether $r$ is positive or negative; it shrinks slowly when $r$

A second derivation, without $S_n$, shows why the formula has this shape. Write $S = u_1 + u_1r + u_1r^{2} + \cdots$

Seeing the partial sums converge

The graph shows the partial sums of two series with first term 6: one with $r = 0.5$, whose sum to infinity is $\frac{6}{1 - 0.5} = 12$, and one with $r = -0.5$

With a positive ratio every term is positive, so the partial sums increase steadily towards $S_\infty$ and never reach it. With a negative ratio the terms alternate in sign, so the partial sums overshoot and undershoot by smaller and smaller amounts. In both cases $S_\infty$ is a value approached, not a value reached after some finite number of terms.

Recurring decimals

A recurring decimal is an infinite geometric series in disguise, so the sum to infinity converts it to a fraction. The decimal $0.272727\ldots$ is

with $u_1 = 0.27$ and $r = 0.01$. The sum is $\frac{0.27}{1 - 0.01} = \frac{0.27}{0.99} = \frac{27}{99} = \frac{3}{11}$

When the sum exists: the condition |r| < 1

Every possible ratio

The table of powers covers all the cases. Using $|r|$, the absolute value of $r$, each case can be stated compactly.

So a geometric series with $u_1 \ne 0$ converges if and only if $|r| < 1$, which is the same as $-1 < r < 1$

Finding the values of a parameter for which the sum exists

When the ratio contains an unknown, the condition $|r| < 1$ becomes an inequality to solve. The absolute value inequality $|r| < 1$ is rewritten as the double inequality $-1 < r < 1$, and the same operations are applied to all three parts.

For the series $5 + \frac{5(x + 1)}{3} + \frac{5(x + 1)^{2}}{9} + \cdots$

The sum to infinity exists only for $-4 < x < 2$. The end points are excluded: at $x = 2$ the ratio is exactly 1, and at $x = -4$ it is exactly $-1$

Working backwards from a sum to infinity

Finding the first term or the ratio

The formula $S_\infty = \frac{u_1}{1 - r}$ links three quantities, so any two give the third. Rearranging gives $u_1 = S_\infty(1 - r)$

For a series with $u_1 = 3$ and $S_\infty = 60$, the ratio is $r = 1 - \frac{3}{60} = 0.95$

Two unknowns: setting up two equations

Often $S_\infty$ is known together with one other fact, so that $u_1$ and $r$ are both unknown. The method is to write one equation from each fact, eliminate $u_1$

When one of the solutions has $|r| \ge 1$, it is rejected, because a series with that ratio has no sum to infinity and so cannot have the given sum. The reason for rejecting it belongs in the written solution, next to the value rejected.

How close is close enough

Because a partial sum never reaches $S_\infty$ when every term is positive, a common task is to find the least number of terms for which $S_n$ exceeds a given value or comes within a given amount of $S_\infty$

Case study: the ITF tennis ball bounce test

The test and the model

The International Tennis Federation tests a ball's bounce by dropping it from a height of 254 cm onto concrete. An approved ball must rebound to a height between 135 cm and 147 cm, so each rebound reaches between $\frac{135}{254} \approx 0.53$ and $\frac{147}{254} \approx 0.58$

Source: Wikipedia, "Tennis ball" (ITF bounce test: dropped from 254 cm, acceptable rebound 135–147 cm at sea level and 20 °C), 2026.

The total vertical distance travelled by the ball if it bounced forever is not simply a single geometric series. The first drop of 254 cm is travelled once, but every rebound height is travelled twice, once up and once down.

Total distance

The rebound heights form an infinite geometric series with $u_1 = 254 \times 0.55 = 139.7$ and $r = 0.55$. Its sum is $\frac{139.7}{1 - 0.55} = 310.4\ldots$

In general, a ball dropped from height $h$ whose rebounds are a fraction $r$ of the previous height travels $h + \frac{2hr}{1 - r} = \frac{h(1 + r)}{1 - r}$

Infinitely many bounces in a finite time

The time taken by the bounces is also a geometric series. A ball falling from rest through a height $H$ takes $\sqrt{\frac{2H}{g}}$

The model therefore predicts infinitely many bounces, all finished within about 5 seconds. This is the same structure as Zeno's dichotomy, with a ball in place of the runner.

Validity of the model

A real ball makes only a limited number of visible bounces before it rolls or rests. The constant ratio is an assumption: at low heights air resistance, the deformation of the ball and the surface change the ratio, and once a bounce is smaller than the ball itself the idea of a "rebound height" stops making sense. Even so, the terms beyond the first ten or so bounces add very little: after 10 rebounds the remaining distance is $2 \times \frac{139.7 \times 0.55^{10}}{0.45} \approx 1.6$ cm. The sum to infinity is therefore a good estimate of the total distance even though the model's infinitely many bounces are not real.

Further applications

Repeated spending: the multiplier effect

When a government spends money, the people who receive it spend a fraction of it, the marginal propensity to consume (MPC), and save the rest. The money they spend is received by others, who again spend the same fraction, and so on. With an initial spend of $10 million and a hypothetical MPC of 0.8, the rounds of spending are $10 million, $8 million, $6.4 million, …, an infinite geometric series with $r = 0.8$.

The ratio $\frac{1}{1 - \text{MPC}} = 5$ is called the multiplier: every dollar of initial spending eventually produces five dollars of total spending in this model. Real economies lose part of each round to imports and taxes, which lowers the ratio and so the multiplier, and the rounds take time, so the infinite total is a long-run upper estimate rather than a forecast for one year.

Medicine doses building up to a steady level

A patient takes a dose of a medicine at the same time each day, and by the next dose the body has removed a fixed fraction of what was there. With a hypothetical dose of 200 mg and 40% of the amount remaining after a day, the amount just after the $n$th dose is

a finite geometric series. The amounts just after each dose are 200, 280, 312, 324.8, 329.9, … mg, and they approach the sum to infinity $\frac{200}{1 - 0.4} \approx 333$ mg. The sum to infinity gives the long-term peak level, which doctors must keep below a harmful level; the speed of approach depends on the fraction remaining.

Fractals: the Koch snowflake

The Koch snowflake starts with an equilateral triangle. At each stage, the middle third of every side is replaced by two sides of a smaller equilateral triangle pointing outwards. The process is repeated forever, creating a fractal, a shape with detail at every scale (subtopic 3.9).

The area is an infinite geometric series. Let the original triangle have area $A$ and side $s$. At stage 1, three new triangles are added, each with side $\frac{s}{3}$ and so with area $\frac{A}{9}$

The perimeter behaves differently. Each stage replaces every side by four sides a third as long, so the perimeter is multiplied by $\frac{4}{3}$ each time: $3s, 4s, \frac{16s}{3}, \ldots$

Source: Wikipedia, "Koch snowflake" (area converges to 8/5 of the original triangle; perimeter infinite), 2026.

Exam focus

Infinite geometric series appear on HL Papers 1 and 2, usually as 2–6-mark parts combined with finite geometric sums from subtopic 1.3 (a term, a finite sum, the least $n$). The command terms are find, determine whether, justify and show that. The recurring shapes are: finding $S_\infty$ for a context and comparing it with a target ("will the balloon ever reach…"); finding $r$ from $u_1$

Part (e) shows why early rounding is dangerous. $S_{16} = 19\,007.6$ is only 0.4 below the target of 19,008, so a ratio or partial sum rounded too soon can give the wrong integer.

The formula itself is rarely the difficulty. Examiners have reported candidates who did not attempt these questions or did not know the formula, even though it is printed in the formula booklet under the HL section of topic 1.

Practice questions

1. Find the sum to infinity of the geometric series $40 + 10 + 2.5 + \cdots$. [2]

2. Determine whether the geometric series $7 + 8.4 + 10.08 + \cdots$ has a sum to infinity. Justify your answer. [2]

3. Use an infinite geometric series to

Summary

• An infinite series converges if its partial sums $S_n$ approach a finite limit, which is its sum; otherwise it diverges.

• For a geometric series, $S_n = \frac{u_1(1 - r^{n})}{1 - r}$

Self-check

• To do:

  I can write the first few partial sums of a series and say whether they appear to converge.

• To do:

  I can explain why $r^{n}$ approaches 0 when $|r| < 1$ and why that gives $S_\infty = \frac{u_1}{1 - r}$