Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Determine if an infinite geometric series converges or diverges.
• Calculate the sum to infinity of a convergent geometric series.
• Use sigma notation to represent and evaluate series.
• Solve application problems with convergent series.
• Distinguish between arithmetic and geometric series.

Lesson summary

This week's focus is on extending our understanding of patterns, sequences, and series. Building on the arithmetic and geometric sequences learned previously, we will delve deeper into infinite geometric series, their convergence and divergence, and sigma notation as a powerful tool to represent series. Understanding these concepts is crucial for modelling various real-world phenomena, from calculating loan repayments and compound interest to predicting population growth and understanding recurring patterns in nature.

Lesson notes

2.1 Infinite Geometric Series: Convergence and Divergence An infinite geometric series is a series with an infinite number of terms, where each term is multiplied by a constant ratio to obtain the next term.

The general form is: a + ar + ar 2 + ar 3 + ...

Where: a is the first term r is the common ratio The crucial question is: When does this series have a finite sum, even though there are infinite terms? The answer lies in the common ratio r.

Convergence: An infinite geometric series converges (has a finite sum) if and only if the absolute value of the common ratio is less than 1: | r | n approaches 0, making the terms of the series become progressively smaller, contributing less and less to the overall sum.

Divergence: An infinite geometric series diverges (does not have a finite sum) if | r | ≥

1. In this case, the terms either stay the same size (if r = 1 or r = -1) or grow larger and larger in magnitude as n approaches infinity.

Example 1: Determining Convergence/Divergence Consider the series: 2 + 1 + 1/2 + 1/4 + ... Here, a = 2 and r = 1/

2. Since |1/2| 1, the series diverges. 2.2 Sum to Infinity (S ∞ ) If an infinite geometric series converges, we can calculate its sum to infinity (S ∞ ) using the following formula: S ∞ = a / (1 - r), where | r | ∞ = 2 / (1 - 1/2) = 2 / (1/2) = 4 Therefore, the sum to infinity of this series is

4. Example 3: Application of Sum to Infinity A rubber ball is dropped from a height of 10 meters. Each time it hits the ground, it bounces to a height that is 3/4 of the previous height. Calculate the total vertical distance travelled by the ball.

Initial drop: 10m First bounce: 10 * (3/4) = 7.5m (up)

First bounce: 7.5m (down)

Second bounce: 7.5 * (3/4) = 5.625m (up)

Second bounce: 5.625m (down) Notice that all bounces except the first drop are covered both up and down.

Therefore, the total distance is: 10 + 2[7.5 + 5.625 + ...] The expression in the brackets is a geometric series with a=7.5 and r=3/4 S ∞ = 7.5 / (1-3/4) = 7.5/(1/4) = 30 Total distance: 10 + 2 * 30 = 70 meters 2.3 Sigma Notation (Σ) Sigma notation provides a concise way to represent a series. The Greek letter sigma (Σ) indicates summation.

General form: ∑ i=m n f(i)

Where: Σ is the summation symbol i is the index of summation (can be any variable) m is the lower limit of summation (the starting value of i) n is the upper limit of summation (the ending value of i) f(i) is the expression being summed, which depends on i. This notation means "the sum of f(i) as i goes from m to n".

Example 4: Expanding Sigma Notation Expand the following sigma notation: ∑ k=1 4 (2k + 1) This means we need to substitute k = 1, 2, 3, and 4 into the expression (2k + 1) and then add the results: (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + (2(4) + 1) = 3 + 5 + 7 + 9 = 24 Example 5: Representing a Series with Sigma Notation Represent the following series using sigma notation: 1 + 4 + 9 + 16 + 25 We recognize that these are perfect squares: 1 2 , 2 2 , 3 2 , 4 2 , 5 2 Therefore, we can write the series as: ∑ i=1 5 i 2 Example 6: Geometric series in Sigma Notation Represent the following series using sigma notation: 3 + 6 + 12 + 24 + 48 This is a geometric series with a = 3 and r =

2. The general term is therefore 3 * 2 (n-1) If we want to sum these first 5 terms, the sigma notation would be: ∑ n=1 5 3 * 2 (n-1) Guided Practice (With Solutions)

Question 1: Determine whether the infinite geometric series 5 + 5/3 + 5/9 + 5/27 + ... converges or diverges. If it converges, calculate its sum to infinity.

Solution: a = 5 r = (5/3) / 5 = 1/3 Since |1/3| ∞ = a / (1 - r) = 5 / (1 - 1/3) = 5 / (2/3) = 5 * (3/2) = 15/2 = 7.5

Commentary: First, we identified a and r. Then we checked the condition for convergence. Finally, we applied the formula for the sum to infinity.

Question 2: Write the series 4 + 8 + 16 + 32 + 64 using sigma notation.

Solution: This is a geometric series with a = 4 and r =

2. The general term is 4 * 2 (n-1) Since there are 5 terms, the sigma notation is: ∑ n=1 5 4 * 2 (n-1)

Commentary: Recognizing the geometric pattern and the number of terms is key.

Question 3: Evaluate the following sigma notation: ∑ i=2 5 (i 2 - 1)

Solution: (2 2 - 1) + (3 2 - 1) + (4 2 - 1) + (5 2 - 1) = (4 - 1) + (9 - 1) + (16 - 1) + (25 - 1) = 3 + 8 + 15 + 24 = 50

Commentary: Substitute each value of i within the given range and sum the results.

Question 4: A company deposits R10,000 into an account that earns 8% interest compounded annually. At the beginning of each year, they deposit an additional R

1

0

0

0. What is the total value of the investment after 5 years (including the initial investment)?

Solution: This requires understanding compound interest and series. We can consider each investment as a separate geometric series.