Infinite Sums Lesson Plan Infinite Sums Lesson Plan
Grade 11 Mathematics • Geometric Series & Limits
Duration: 45-50 min
Learning Objective
Students will derive the formula for the sum of an infinite geometric series by evaluating the limit of the finite sum formula as \(n \to \infty\), specifically for ratios where \(|r| < 1\).
Materials
Infinite Limits Presentation Slides
Summation Secrets Worksheet
Calculators
YouTube Video: "Infinite Geometric Series - How To Calculate..."
Key Concepts
Limits at Infinity (\(\lim_{n \to \infty}\))
Convergence vs. Divergence
Exponential Decay (\(|r| < 1\))
Lesson Flow
0-5 MIN
Warm-up: The Vanishing Term
Students calculate \(0.5^2\), \(0.5^{10}\), and \(0.5^{100}\). Teacher Prompt: "As the exponent grows, what is the 'destination' of this number? Does it ever reach zero? Does it ever go negative?"
5-15 MIN
Direct Instruction: Visualizing Infinity
Play the provided video. Focus specifically on [6:45-8:29] for the derivation. Encourage students to sketch the "square subdivision" visual in their notes to connect the algebra to geometry.
15-30 MIN
Derivation Workshop: The "Why"
Students work in pairs on the "Derivation Guide" section of the worksheet. They must explicitly use the limit notation: \(\lim_{n \to \infty} r^n = 0\). Scaffold: If students struggle, ask them to plug in \(r=2\) and see why the sum "explodes."
30-35 MIN
Extension: The 0.999... Paradox
Students apply the formula to \(0.9 + 0.09 + 0.009...\). This provides a concrete, mind-blowing application of infinite sums proving that \(0.999... = 1\).
Teacher Tips & Differentiation
Common Misconceptions
Students often apply the formula to divergent series (where \(|r| \ge 1\)). Emphasize that the "vanishing term" logic ONLY works if the ratio is shrinking the values.
Support & Extension
Support: Use a physical piece of paper and keep tearing it in half to demonstrate \((1/2)^n \to 0\).Extension: Challenge students to find the sum of \(1 + 1/2 + 1/4...\) using only a 1x2 rectangle.
Infinite Sums Slides INFINITE SUMS
Deriving the formula for geometric series that never end.
Grade 11 Mathematics Limits & Convergence
WARM-UP
The Vanishing Power
Calculate
\(0.5^2\)
Calculate
\(0.5^{10}\)
Calculate
\(0.5^{100}\)
"What happens as the exponent gets bigger and bigger?"
What did we observe?
1
The values get smaller as the exponent increases.
2
The number approaches zero but never actually reaches it.
3
We call this a Limit.
\(\lim_{n \to \infty} 0.5^n = 0\)
"As n goes to infinity, 0.5 to the power of n approaches zero."
The Derivation [6:45 - 8:29]
Embedded media
Pay close attention to how the finite formula changes!
Starting Point: Finite Sum
\(S_n = \frac{a(1 - r^n)}{1 - r}\)
\(a\) First Term
\(r\) Common Ratio
\(n\) Number of Terms
Workshop Challenge
In pairs, look at the finite formula. Explain what happens to the \(r^n\) term as \(n \to \infty\) in two different scenarios:
Scenario A
\(|r| < 1\)
(e.g., \(r = 0.5\))
Scenario B
\(|r| > 1\)
(e.g., \(r = 2.0\))
The Infinite Sum Formula
Only when \(|r| < 1\), the \(r^n\) term vanishes to zero.
\(S_\infty = \frac{a}{1 - r}\)
The \(0.999...\) Paradox
Is \(0.999...\) actually equal to \(1\)?
We can write it as a series:
\(0.9 + 0.09 + 0.009 + ...\)
• What is the first term (\(a\))?
• What is the common ratio (\(r\))?
• Apply the formula!
Summation Secrets Worksheet Summation Secrets
Deriving the Infinite Geometric Series Formula
Student:
Date:
01 Warm-up: The Limit of Decay
\(0.5^2\)
\(0.5^{10}\)
\(0.5^{100}\)
Observation: What is the "destination" of these values?
02 Visualizing Infinity
Sketch the subdivision visual
As \(n \to \infty\), if \(|r| < 1\), the term \(r^n\) approaches:
03 Derivation Workshop
Finite Sum Formula: \(S_n = \frac{a(1 - r^n)}{1 - r}\)
Paragraph: Why does the formula simplify when \(|r| < 1\)?
Infinite Formula:
\(S_\infty = \)
04 The \(0.999...\) Challenge
Series: \(0.9 + 0.09 + 0.009 + ...\)
a =
r =
Show Work / Calculation:
Summation Secrets Answer Key Summation Secrets
TEACHER ANSWER KEY
Geometry & Limits
01 Warm-up: The Limit of Decay
\(0.5^2\)
0.25
\(0.5^{10}\)
0.000976...
\(0.5^{100}\)
Approaches 0
Observation: What is the "destination" of these values?
The values get exponentially smaller. The destination is zero. This illustrates that \(0.5^\infty\) effectively equals 0.
02 Visualizing Infinity
Expect a fractal sketch
As \(n \to \infty\), if \(|r| < 1\), the term \(r^n\) approaches:
0
03 Derivation Workshop
Finite Sum Formula: \(S_n = \frac{a(1 - r^n)}{1 - r}\)
Paragraph Explanation:
"When \(|r| < 1\), the term \(r^n\) represents exponential decay. As \(n\) goes to infinity, the limit of \(r^n\) is 0. This means the \((1 - r^n)\) part of the numerator becomes \((1 - 0)\), which is just 1. This simplifies the whole expression to just \(a / (1-r)\)."
Infinite Formula:
\(S_\infty = \frac{a}{1 - r}\)
04 The \(0.999...\) Challenge
Series: \(0.9 + 0.09 + 0.009 + ...\)
a = 0.9
r = 0.1
\(S_\infty = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1\)
Hence, 0.999... = 1
