Sum to Infinity | Convergence & Applications

Learning Objectives

By the end of this article, you should be able to:

• Identify the convergence condition \( |r| < 1 \) for an infinite geometric series.
• Calculate the sum to infinity \( S_\infty \) using the formula.
• Convert recurring decimals to fractions using infinite geometric series.
• Solve problems involving infinite processes (e.g., bouncing balls, distance traveled).
• Apply sum to infinity to financial and other real-world contexts.
• Distinguish between finite and infinite sums and choose the correct formula.

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1. Introduction:

Sum to infinity appears every year in DSE Paper 1 Section A(2) and Section B. Questions may ask you to:

• Find the sum to infinity of a geometric series
• Determine whether a series converges
• Convert a recurring decimal to a fraction
• Solve problems involving infinite geometric processes

These are high-level skills that can help you secure top marks in the exam.

2. Convergence Condition

An infinite geometric series \( a + ar + ar^2 + ar^3 + \cdots \) converges (has a finite sum) only if:

If \( |r| \ge 1 \), the series diverges (does not have a finite sum).

Examples

• \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \) converges (\( r = \frac{1}{2} \))
• \( 3 + 6 + 12 + 24 + \cdots \) diverges (\( r = 2 \))
• \( 5 - 5 + 5 - 5 + \cdots \) diverges (\( r = -1 \))

3. Sum to Infinity Formula

When \( |r| < 1 \), the sum to infinity is given by:

This is derived from the finite sum formula by taking \( n \to \infty \), since \( r^n \to 0 \) when \( |r| < 1 \).

Worked Example

Question: Find the sum to infinity of \( 8 + 4 + 2 + 1 + \cdots \).

4. Converting Recurring Decimals to Fractions

A recurring decimal (e.g., \( 0.\overline{3} \)) can be expressed as an infinite geometric series and converted to a fraction.

Method

1. Write the decimal as a sum of its repeating digits.
2. Identify the first term \( a \) and the common ratio \( r \).
3. Use the formula \( S_\infty = \frac{a}{1-r} \).

Worked Example

Question: Convert \( 0.\overline{36} = 0.363636\ldots \) to a fraction.

5. Applications of Sum to Infinity

Sum to infinity is used to model situations involving infinite repetition, such as:

• Bouncing balls: Total distance traveled
• Infinite processes: Repeated actions with decreasing effect
• Financial models: Perpetuities (infinite annuity) in finance
• Fractal geometry: Infinite perimeter/area

Worked Example (Bouncing Ball)

Question: A ball is dropped from a height of 10 m. After each bounce, it rebounds to \( \frac{3}{4} \) of its previous height. Find the total vertical distance traveled by the ball until it comes to rest.

6. Worked Examples

Example 1: Basic Sum to Infinity

Question: Find the sum to infinity of \( 12 - 6 + 3 - 1.5 + \cdots \).

Example 2: Recurring Decimal

Question: Express \( 2.\overline{5} = 2.555\ldots \) as a fraction.

Example 3: Application

Question: A person walks 20 m east, then turns around and walks 10 m west, then 5 m east, then 2.5 m west, and so on. What is the total distance from the starting point?

7. DSE-Style Practice Questions

8. Solutions with Explanations

9. Exercise

Click the following link to have
 An Exercise on Sum to Infinity | Convergence & Applications

Key Takeaways

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Summary Checklist for Revision

• Convergence condition: \( |r| < 1 \)
• Sum to infinity: \( S_\infty = \frac{a}{1-r} \)
• Recurring decimal conversion: \( 0.\overline{ab} = \frac{ab}{99} \)
• Bouncing ball: total = \( h + 2 \times \frac{hr}{1-r} \)
• Infinite series applications
• Distinguish finite vs infinite sums
• Check if \( r \) is within range