Mathematics · Year 13

Active learning ideas

Sum to Infinity of a Convergent Geometric Series

Challenge your students with the mind-bending idea of infinity by asking: can you add up an infinite list of numbers and get a finite answer?

• National Curriculum Attainment Targets
• DfE Subject Content for Mathematics: D3 - Understand and use the condition for the convergence of a geometric series, and the formula for the sum to infinity of a convergent geometric series.

• 15–25 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Collaborative Problem-Solving
• 25 min · Small Groups

The Bouncing Ball Problem

In small groups, students are given the initial drop height of a ball and a coefficient of restitution (the 'bounciness' factor, which serves as the common ratio). They calculate the total vertical distance the ball travels by summing the infinite series of upward and downward movements.

Justify the condition |r| < 1 for the convergence of a geometric series.

Facilitation Tip

Encourage students to draw a diagram to visualise the separate 'up' and 'down' journeys after the initial drop.

What to look for

Use mini-whiteboards for a quick-fire round where students are shown a series and must write down the value of 'r' and state 'converges' or 'diverges'.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 20 min · Pairs

Zeno's Paradox Investigation

Present Zeno's dichotomy paradox (to travel a distance, one must first travel half, then half the remaining distance, and so on). Students model the journey as a geometric series and use the sum to infinity formula to show that the traveller does, in fact, reach their destination.

Explain the concept of a limit in the context of an infinite geometric series.

Facilitation Tip

Use a number line on the board to visually represent the first few terms of the series to build intuition.

What to look for

An exam-style question that requires students to find the sum to infinity of a numeric series, then find the range of values of x for which an algebraic series converges, and finally apply the concept to a contextual problem.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 15 min · Individual

Decimal to Fraction Conversion

Challenge students to express a recurring decimal, such as 0.777... or 0.212121..., as an infinite geometric series. They then use the sum to infinity formula to find its equivalent fractional form.

Analyse a real-world problem, such as the total distance travelled by a bouncing ball, using the sum to infinity formula.

Facilitation Tip

Start with a simple example like 0.333... to help students identify the first term 'a' and the common ratio 'r'.

What to look for

Provide students with a checklist of skills (e.g., 'I can find 'r'', 'I can test for convergence', 'I can use the S_∞ formula') for them to traffic-light their confidence levels.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start by visually demonstrating a convergent series, such as 1/2 + 1/4 + 1/8 + ..., by shading parts of a square to show it approaches 1. Explicitly link the behaviour of r^n as n → ∞ to the condition |r| < 1. Insist that students write down the check for convergence as the first step in any problem before they apply the formula S_∞ = a / (1 - r).

By the end of this topic, students will be able to confidently determine if a geometric series converges and calculate its sum to infinity to solve a variety of pure and applied problems.

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Watch Out for These Misconceptions

• Any infinite series can have its sum calculated with the formula.

  The formula S_∞ = a / (1 - r) is only valid for convergent geometric series where the magnitude of the common ratio, |r|, is less than 1. If |r| ≥ 1, the series diverges and its sum tends towards infinity (or does not approach a single value).

• Adding an infinite number of positive terms must result in an infinite sum.

  This is a conceptual hurdle. Explain that if the terms are getting progressively smaller (which happens when |r| < 1), the amount being added each time decreases. The sum approaches a finite 'limit' or boundary that it never crosses, much like how you can keep adding half the remaining distance to a wall but never travel more than the initial distance.

• In the condition |r| < 1, the modulus sign is unimportant.

  The modulus is critical. A series with r = -0.5 converges, as |-0.5| < 1. A series with r = -2 diverges, as |-2| > 1. The modulus ensures we consider the magnitude of the ratio, as this determines whether the terms shrink towards zero.

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Methods used in this brief

• Collaborative Problem-Solving