Infinity Bound Slides Infinity Bound
Approaching the Edge of Sequences
Lesson 1: Introduction to Limits
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The Tortoise and the Hare
Zeno's Paradox: To reach the finish line, you must first travel half the distance.
To travel the remaining half, you must travel half of that distance.
"If you keep dividing the distance, will you ever actually arrive?"
1/2 + 1/4 + 1/8 + 1/16 + ...
The Mathematical Limit
\[ \lim_{n \to \infty} a_n = L \]
"The limit of the sequence \( a_n \) as \( n \) approaches infinity is \( L \)."
Convergence
The terms of the sequence get closer and closer to a single, finite number.
Divergence
The terms do not approach a finite number (they grow without bound or oscillate).
Graphing Convergence
Sequence A: \( a_n = \frac{1}{n} \)
Terms: 1, 0.5, 0.33, 0.25, 0.2, ...
Limit: 0 (Converges)
Sequence B: \( a_n = 2n + 1 \)
Terms: 3, 5, 7, 9, 11, ...
Limit: \( \infty \) (Diverges)
Approaching zero...
Quick Check
Does this sequence converge or diverge?
\( a_n = (-1)^n \)
A) Converge
B) Diverge
Limit Hunters Worksheet Limit Hunters
Analyzing Sequence End-Behavior
Name:
Date:
Part 1: The Paradox
Recall Zeno's Paradox of the tortoise. If the distance to the finish line is 1 unit, and you always travel half the remaining distance, the sequence of total distance traveled is: 1/2, 3/4, 7/8, 15/16, ...
1. Write the next two terms of this sequence.
2. Does this sequence approach a specific finite number? If so, what is the limit?
Part 2: Hunting for Limits
Calculate the first four terms of each sequence, then determine if it converges or diverges . If it converges, state the limit.
\( a_n = \frac{n}{n+1} \) Sequence A
n=1
n=2
n=3
n=4
Convergence/Divergence
Limit Value (if any)
\( a_n = 3 + (-1)^n \) Sequence B
n=1
n=2
n=3
n=4
Convergence/Divergence
Limit Value (if any)
Part 3: The Big Picture
"If a sequence is bounded (it never goes above or below certain values) and always moves in one direction (increasing or decreasing), it MUST converge."
Think about a sequence where you keep adding grains of sand to a pile, but each time you add half as much as the time before. Use the statement above to explain why the height of the pile cannot grow forever.
Limit Hunters Answer Key Answer Key
Limit Hunters Worksheet
Teacher Resource
Part 1: The Paradox
1. Next two terms:
31/32, 63/64
2. Approach finite number? Limit?
Yes, the sequence approaches 1. As the numerator gets closer to the denominator, the value gets closer to 1 but never exceeds it.
Part 2: Hunting for Limits
Sequence A: \( a_n = \frac{n}{n+1} \)
Terms: 1/2, 2/3, 3/4, 4/5 (or 0.5, 0.67, 0.75, 0.8)
Convergence: CONVERGES
Limit: L = 1
Sequence B: \( a_n = 3 + (-1)^n \)
Terms: 2, 4, 2, 4
Convergence: DIVERGES (Oscillates between 2 and 4)
Limit: NONE
Part 3: The Big Picture
Explanation of the sand pile:
The pile cannot grow forever because the total height is bounded . Since you only add half as much sand as before, the total added amount follows the series \( 1/2 + 1/4 + 1/8... \), which we know approaches but never exceeds 1. Because the sequence of heights is increasing but also bounded above by 1 (plus the initial pile height), the height must converge to a specific finite value rather than growing infinitely.
Sigma Notation Slides The Power of Sigma
Elegant Notation for Infinite Sums
Lesson 2: Summation Notation
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The Problem with Dots
The "Old" Way:
1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + ... + 10,000
The Sigma Way:
\[ \sum_{n=1}^{100} n^2 \]
Anatomy of a Sum
\[ \sum_{i=1}^{k} a_i \]
Upper Limit (Where to stop)
Lower Limit / Index (Where to start)
The Argument (The formula)
The Big Shift
1
Sequence
A comma-separated list of values.
2, 4, 6, 8, 10
2
Series
The sum of those values.
2 + 4 + 6 + 8 + 10
"Sequences are lists. Series are sums."
Challenge: Expand & Solve
\[ \sum_{j=2}^{5} (2j - 1) \]
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=
Final Sum: ___
Sigma Cipher Worksheet Sigma Cipher
Decoding and Encoding Summation Notation
Name:
Date:
Part 1: The Decoder
Expand each sigma notation into its sum form and calculate the final result.
\[ \sum_{n=1}^{5} (2n) \]
Expanded:
Sum:
\[ \sum_{k=0}^{3} (k^2 + 1) \]
Expanded:
Sum:
Part 2: The Encoder
Write the following series using Sigma notation. Find the pattern first!
5 + 10 + 15 + 20 + 25
Your Notation
\[ \sum \]
1 + 3 + 9 + 27 + 81
Your Notation
\[ \sum \]
The Infinity Challenge
Consider the following notation:
\[ \sum_{i=1}^{\infty} \frac{1}{2^i} \]
Write out the first four terms of this series. What do you think happens as you add more and more terms forever?
Sigma Cipher Answer Key Answer Key
Sigma Cipher Worksheet
Teacher Resource
Part 1: The Decoder
\[ \sum_{n=1}^{5} (2n) \]
Expanded: 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2 + 4 + 6 + 8 + 10
Sum: 30
\[ \sum_{k=0}^{3} (k^2 + 1) \]
Expanded: (0²+1) + (1²+1) + (2²+1) + (3²+1) = 1 + 2 + 5 + 10
Sum: 18
Part 2: The Encoder
5 + 10 + 15 + 20 + 25
\[ \sum_{n=1}^{5} 5n \]
(Or equivalent notation starting at a different index)
1 + 3 + 9 + 27 + 81
\[ \sum_{n=0}^{4} 3^n \]
(Or equivalent notation)
Part 3: The Infinity Challenge
Terms: 1/2, 1/4, 1/8, 1/16
Observation: As more terms are added, the sum gets closer and closer to 1. Even with an infinite number of terms, the sum will never exceed 1. This is a convergent infinite series with a limit of 1.
Partial Sum Formulas Slides Adding It All Up
The Genius of Partial Sums
Lesson 3: Arithmetic and Geometric Series
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The Legend of Gauss
In the 1780s, a schoolmaster gave his class a "busy work" assignment:
"Add every number from 1 to 100."
Young Carl Friedrich Gauss put his slate down with the correct answer in seconds.
How did he do it?
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
... and so on ...
50 pairs of 101
Arithmetic Formula
The Sum of \( n \) terms
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
n
Number of terms
\( a_1 \)
The first term
\( a_n \)
The last term
Geometric Formula
When terms grow by a ratio rather than a common difference, we use:
\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]
r = common ratio
n = number of terms
Stacking the Numbers
Think of partial sums as accumulations.
Application
How many blocks are in a pyramid that is 50 levels high, if the top layer has 1 block and each layer below has 1 more block than the one above it?
...
LAYER 50
Summation Showdown Worksheet Summation Showdown
Arithmetic and Geometric Partial Sums
Name:
Date:
Arithmetic Sum
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Geometric Sum
\[ S_n = \frac{a_1(1 - r^n)}{1 - r} \]
Part 1: The Gauss Challenge
1. Find the sum of the first 50 positive even integers: 2 + 4 + 6 + ... + 100
n
a₁
aₙ
Show work here
2. A theater has 20 rows of seats. The first row has 15 seats, and each subsequent row has 2 more seats than the row before it. How many total seats are in the theater?
Part 2: Rapid Growth
3. Calculate the sum of the first 8 terms of the geometric series: 3 + 6 + 12 + ...
a₁
r
n
Show work here
4. The Viral Video Paradox: On Day 1, a video gets 1,000 views. Each day, the number of views increases by 50% (r = 1.5). What is the total number of views after 7 days?
Summation Showdown Answer Key Answer Key
Summation Showdown Worksheet
Teacher Resource
Part 1: Arithmetic Sums
1. Sum of first 50 even integers (2 to 100)
n: 50
a₁: 2
aₙ: 100
\[ S_{50} = \frac{50}{2}(2 + 100) = 25(102) = \mathbf{2,550} \]
2. Theater seating (20 rows, start at 15, d=2)
Find last row: \( a_{20} = 15 + (20-1)2 = 15 + 38 = 53 \)
Find sum: \( S_{20} = \frac{20}{2}(15 + 53) = 10(68) = \mathbf{680} \text{ seats} \)
Part 2: Geometric Sums
3. Sum of first 8 terms (3, 6, 12...)
a₁: 3
r: 2
n: 8
\[ S_8 = \frac{3(1 - 2^8)}{1 - 2} = \frac{3(1 - 256)}{-1} = \frac{-765}{-1} = \mathbf{765} \]
4. Viral Video (a₁=1000, r=1.5, n=7)
\[ S_7 = \frac{1000(1 - 1.5^7)}{1 - 1.5} = \frac{1000(1 - 17.0859375)}{-0.5} \]
Total views: \(\approx \mathbf{32,172} \text{ views} \)
Infinite Sums Slides Summing the Unending
When Infinite Becomes Finite
Lesson 4: Infinite Geometric Series
The Halving Experiment
Imagine a piece of string 1 unit long.
Cut it in half. Lay the piece down.
Cut the remaining piece in half again. Add it.
Repeat this process forever.
"Can you ever lay down more than 1 unit of string?"
\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1 \]
The Convergence Rule
An infinite geometric series only has a finite sum if the terms shrink fast enough to zero.
Converges If
\[ |r| < 1 \]
The "Sweet Spot"
Diverges If
\[ |r| \ge 1 \]
Grows Without Bound
The Infinite Formula
The Sum of All Terms
\[ S = \frac{a_1}{1 - r} \]
\( a_1 \)
The First Term
\( r \)
The Common Ratio
Mystery of the Repeating Nine
Does \( 0.999... \) exactly equal 1?
\[ 0.9 + 0.09 + 0.009 + \dots \]
First term \( a_1 = 0.9 \)
Ratio \( r = 0.1 \)
The Proof
\[ S = \frac{0.9}{1 - 0.1} \]
\[ S = \frac{0.9}{0.9} = 1 \]
Mind = Blown
Infinite Sums Investigation Worksheet Infinite Investigations
When Sums Settle Down
Name:
Date:
Part 1: The Gateway
Determine if each infinite geometric series converges or diverges by identifying the ratio \( r \).
\( \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \dots \)
Ratio (r)
Converge/Diverge?
\( 2 + 3 + 4.5 + 6.75 + \dots \)
Ratio (r)
Converge/Diverge?
Part 2: Finding the Limit
Use the formula \( S = \frac{a_1}{1 - r} \) to find the sum of the convergent series below.
3. \( 10 + 5 + 2.5 + 1.25 + \dots \)
Show calculations here
4. \( \sum_{n=1}^{\infty} 4(\frac{1}{4})^{n-1} \)
Show calculations here
Part 3: Decimals as Series
Convert the repeating decimal \( 0.\overline{42} \) into a geometric series and then into its simplified fraction form.
Series Form:
First Term (\( a_1 \))
Common Ratio (\( r \))
Final Fraction
___ / ___
Infinite Sums Answer Key Answer Key
Infinite Investigations Worksheet
Teacher Resource
Part 1: The Gateway
Series A: 1/3 + 1/9 + 1/27...
r: 1/3
Status: CONVERGES (|1/3| < 1)
Series B: 2 + 3 + 4.5...
r: 1.5
Status: DIVERGES (|1.5| ≥ 1)
Part 2: Finding the Limit
3. 10 + 5 + 2.5 + ...
a₁ = 10, r = 0.5
\[ S = \frac{10}{1 - 0.5} = \frac{10}{0.5} = \mathbf{20} \]
4. \(\sum_{n=1}^{\infty} 4(\frac{1}{4})^{n-1}\)
a₁ = 4, r = 0.25
\[ S = \frac{4}{1 - 0.25} = \frac{4}{0.75} = \mathbf{\frac{16}{3}} \approx \mathbf{5.33} \]
Part 3: Decimals as Series
Decimal: \( 0.\overline{42} \)
Series Form: \( 0.42 + 0.0042 + 0.000042 + \dots \)
a₁: 0.42
r: 0.01
\[ S = \frac{0.42}{1 - 0.01} = \frac{0.42}{0.99} = \mathbf{\frac{42}{99}} = \mathbf{\frac{14}{33}} \]
Fractal Frontier Slides Fractal Frontiers
The Paradox of Infinite Perimeters
Lesson 5: Fractals and Limits
Self-Similarity
A fractal is a geometric shape that repeats its structure at every scale.
"No matter how much you zoom in, the pattern remains identical."
Nature uses fractals in coastlines, ferns, clouds, and snowflakes.
X
Iterative Rule: Repeat Forever
Iteration 1 to \(\infty\)
Iteration 0
Iteration 1
Iteration 2
Infinity
Perimeter = (Original) \(\times (\frac{4}{3})^n\)
Finite Area, Infinite Perimeter
The Perimeter
Each step increases the length by 4/3. Since \( 4/3 > 1 \), the sequence of perimeters diverges.
\[ \lim_{n \to \infty} P_n = \infty \]
The Area
Each new triangle is smaller than the previous. The total area converges to exactly 1.6 times the original triangle.
\[ \lim_{n \to \infty} A_n = \frac{8}{5}A_0 \]
"You can hold an infinite coastline in the palm of your hand."
Limits in Nature
How long is the coastline of Britain?
The Coastline Paradox states that if you measure with a smaller and smaller ruler, the length keeps growing... just like a fractal!
Discussion
In the real world, is a coastline truly infinite? Why or why not?
Fractal Paradox Lab Worksheet Fractal Paradox Lab
Analyzing the Koch Snowflake
Name:
Date:
Part 1: Unending Edges
To create the Koch Snowflake, we replace the middle third of each line segment with two sides of an equilateral triangle. This increases the perimeter by a factor of 4/3 each time.
Iteration (n) Formula Perimeter (if original = 3) 0 (Triangle) 3 3 1 \( 3 \times (\frac{4}{3}) \) 4 2 \( 3 \times (\frac{4}{3})^2 \) 3 \( 3 \times (\frac{4}{3})^3 \)
Based on the ratio \( r = 4/3 \), use limit notation to describe what happens to the perimeter as the number of iterations reaches infinity.
Part 2: Contained Spaces
While the perimeter grows, each new triangle added is smaller than the last. The total area is the sum of the original triangle plus thousands of tiny new ones.
The sequence of areas is convergent . If the area of the original triangle is 1, the total area after infinite steps is exactly \( \frac{8}{5} = 1.6 \).
The Paradox
Explain in your own words: How can a shape have a "length" that is infinite, but a "surface" that is finite? (Hint: Think about boundaries!)
Your Explanation
Fractal Paradox Answer Key Answer Key
Fractal Paradox Lab Worksheet
Teacher Resource
Part 1: Unending Edges
n Perimeter (a₀ = 3) 0 3 1 4 2 5.33 (or 16/3) 3 7.11 (or 64/9)
Limit Notation Explanation:
\[ \lim_{n \to \infty} P_n = \infty \]. Since the common ratio (4/3) is greater than 1, the geometric sequence of perimeters is divergent and grows without bound.
Part 2: Contained Spaces
Sample Explanation for the Paradox:
The perimeter becomes infinite because we are adding finer and finer "crinkles" to the edge at every scale. However, each new crinkle is contained within a specific region. The area converges because the amount of area added at each step follows a convergent geometric series (the new triangles shrink in area by a factor of 1/9 each iteration). This shows that a boundary can be infinitely long while still enclosing a finite space.
