Limit Horizons Slides Limit Horizons
Visualizing the Infinite Sequence
The Big Question
"How do we mathematically determine the ultimate destination of an infinite process that never actually ends?"
Today's Objectives:
Visualize sequence behavior using plotting
Master limit notation: \(\lim_{n \to \infty} a_n = L\)
Classify sequences: Convergent vs. Divergent
The Discrete Journey
Definition
A sequence \(\{a_n\}\) is a list of numbers in a specific order:
\(a_1, a_2, a_3, \dots, a_n, \dots\)
Think of it as a function whose domain is the set of positive integers.
Example 1: Harmonic
\(a_n = \frac{1}{n} \to 1, \frac{1}{2}, \frac{1}{3}, \dots\)
Example 2: Alternating
\(b_n = (-1)^n \to -1, 1, -1, 1, \dots\)
Three Possible Fates
Convergence
Terms get closer and closer to a single real number \(L\).
Divergence
Terms grow without bound towards \(\infty\) or \(-\infty\).
Oscillation
Terms bounce back and forth, never settling on a single value.
"Convergence is the existence of a destination."
The e Experiment
What happens to the sequence \(a_n = (1 + \frac{1}{n})^n\) as \(n\) becomes massive? Does it explode to infinity? Or does it hit a ceiling?
Let's check the first few steps:
\(n = 1 \implies (1 + 1)^1 = 2\)
\(n = 2 \implies (1 + 0.5)^2 = 2.25\)
\(n = 10 \implies (1.1)^{10} \approx 2.5937\)
Your task: Find out what happens when \(n=10,000\).
Discovery Log Worksheet The \(e\) Experiment
Discovering the Natural Limit through Sequences
Name:
Date:
Introduction
A central question in calculus is understanding behavior at infinity. Today, we investigate one of the most important sequences in mathematics: \(a_n = \left(1 + \frac{1}{n}\right)^n\). As \(n\) increases, the term inside the parentheses gets smaller, but the exponent gets larger. Who wins this tug-of-war?
Part 1: Initial Ascent
Calculate the following values for \(a_n = (1 + \frac{1}{n})^n\). Round to 4 decimal places.
\(n\) Calculation \(a_n\) (Value) 1 \((1 + 1/1)^1\) 2.0000 2 \((1 + 1/2)^2\) 2.2500 5 10 100
Part 2: The Horizon Graph
Based on your table, plot the points on the grid below. What trend do you observe?
n
a_n
Observations:
Part 3: The Infinite Limit
Use a calculator or spreadsheet to find the value of \(a_n\) for \(n = 10,000\).
My Result for \(n = 10,000\):
1. Does the sequence appear to converge, diverge, or oscillate? Explain your reasoning based on the change in growth rate.
2. Using limit notation, write the statement for the limit of this sequence. (Hint: Research the number \(e\)).
\(\lim_{n \to \infty} (1 + \frac{1}{n})^n = \) ________
"The number e is a mathematical constant approximately equal to 2.71828..." — Leonard Euler
Algebraic Engines Slides Algebraic Engines
The Mechanics of Limit Laws
The \(\frac{\infty}{\infty}\) Dilemma
When both numerator and denominator grow without bound, we call it an indeterminate form .
It doesn't mean "no answer."
It means we need more algebraic power to see the hidden destination.
\(\frac{\infty}{\infty}\)
The "Big Term" Strategy
The Dominant Term
In any polynomial, the term with the highest power of \(n\) eventually dictates the entire expression's behavior.
The Algebraic Fix
Divide every term by the highest power of \(n\) found in the denominator .
Evaluate:
\(\lim_{n \to \infty} \frac{3n^2 + 5}{2n^2 - n}\)
Step 1: Divide by \(n^2\)
\( = \lim_{n \to \infty} \frac{3 + \frac{5}{n^2}}{2 - \frac{1}{n}}\)
Step 2: Apply Limit Laws
\( = \frac{3 + 0}{2 - 0} = \frac{3}{2}\)
Shortcut: Degree Analysis
Deg(Num) < Deg(Den)
The denominator grows faster.
Limit = 0
Deg(Num) = Deg(Den)
The growth is balanced.
Ratio of Coeffs
Deg(Num) > Deg(Den)
The numerator explodes.
Divergent (\(\infty\))
Ready for the Limit Race?
You will be given 10 sequences. You have 10 seconds for each to decide:
Destined for 0
Stable at a Constant
Exploding to \(\infty\)
Limit Race Sprint Worksheet Limit Race Sprint
Rational Sequence Speed Trials
Racer:
Time:
Objective
Rapidly determine the limit of each sequence. Use your knowledge of leading terms and degree comparison. You do not need to show full algebraic steps for this sprint, but you must be accurate!
LIMIT CODES
0 : Zero
C : Constant
D : Divergent
01
\(a_n = \frac{5n^2 - 100}{2n^2 + 50n}\)
Limit:
02
\(b_n = \frac{n + 1}{n^2 - 1}\)
Limit:
03
\(c_n = \frac{4n^3 + 2n}{n^2 + 10^6}\)
Limit:
04
\(d_n = \frac{(n+1)(2n-1)}{3n^2}\)
Limit:
05
\(e_n = \frac{\sqrt{4n^2 + 1}}{3n + 2}\)
Limit:
The Final Lap: Showing Your Work
Choose any one sequence from above (01-05) and prove the limit rigorously using the "divide by highest power" method.
Algebraic Engines // Efficiency is Elegance
Squeeze Theorem Slides The Squeeze Theorem
The Art of Mathematical Bounding
The Oscillator
How do we handle sequences like:
\(a_n = \frac{\sin(n)}{n}\)
No algebraic cancellation possible
Sine oscillates between -1 and 1 forever
"The numerator is wild, but the denominator is powerful. To prove the limit is zero, we must 'trap' the function."
The Theorem
If we have three sequences \(\{a_n\}\), \(\{b_n\}\), and \(\{c_n\}\) such that:
\(a_n \leq b_n \leq c_n\)
for all \(n\) beyond some value, and:
\(\lim_{n \to \infty} a_n = L\) and \(\lim_{n \to \infty} c_n = L\)
Then \(\lim_{n \to \infty} b_n = L\)
Visualizing the Trap
Upper Bound \(c_n\)
Oscillator \(b_n\)
Lower Bound \(a_n\)
They all meet at \(L\)!
Your Bounding Toolkit
Trigonometry
\(-1 \leq \sin(n) \leq 1\)
\(-1 \leq \cos(n) \leq 1\)
Use these to create the outer "slices" of the sandwich.
Sign Preservation
Always preserve the inequality sign by multiplying/dividing by positive expressions of \(n\).
\(\frac{-1}{n} \leq \frac{\sin(n)}{n} \leq \frac{1}{n}\)
Squeeze Play Activity Worksheet Squeeze Play Activity
Proving Convergence via Bounding
Section:
Date:
The Squeeze Recipe
Identify the oscillating or "difficult" part of the sequence.
Find a known lower bound (\(a_n\)) and upper bound (\(c_n\)).
Show that both \(\lim_{n \to \infty} a_n\) and \(\lim_{n \to \infty} c_n\) equal the same value \(L\).
Conclude that the target sequence also converges to \(L\).
Problem 1: The Basic Oscillator
Difficulty: Level 1
Prove that the sequence \(b_n = \frac{(-1)^n}{n^2 + 1}\) converges using the Squeeze Theorem.
Step 1: Set the Bounds
_______ \(\leq \frac{(-1)^n}{n^2 + 1} \leq\) _______
Step 2: Take Limits
\(\lim_{n \to \infty} (\text{lower}) = \) ______
\(\lim_{n \to \infty} (\text{upper}) = \) ______
Formal Conclusion:
Problem 2: Trig Trap
Difficulty: Level 2
Evaluate the limit of the sequence: \(a_n = \frac{n \cdot \cos(n^2)}{n^3 + 4}\)
Space for Bounding Argument:
Pro-Tip:
Remember that \(|\cos(x)| \leq 1\) for any value of \(x\). Don't let the squared term \(n^2\) inside the cosine distract you—the range remains the same!
Problem 3: Factorial Growth
Challenge
Use the Squeeze Theorem to show that \(\lim_{n \to \infty} \frac{n!}{n^n} = 0\). (Hint: Write out the terms of the fraction and compare to \(\frac{1}{n}\)).
"The Squeeze Theorem: When you can't hit the target directly, box it in."
Growth Wars Slides Growth Wars
L'Hôpital's Rule for Sequences
The Calculus Connection
Sequences are discrete (dots). L'Hôpital's Rule requires derivatives , which need continuity (lines).
The Bridge Theorem
If \(\lim_{x \to \infty} f(x) = L\), and \(a_n = f(n)\), then \(\lim_{n \to \infty} a_n = L\).
Discrete \(\to\) Continuous
If the line goes there, the dots must follow.
The Heavy Artillery
For indeterminate forms of type \(\frac{\infty}{\infty}\) or \(\frac{0}{0}\):
\(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{f'(x)}{g'(x)}\)
Warning: Differentiate numerator and denominator separately . Do NOT use the quotient rule!
The Hierarchy of Speed
Who grows faster as \(n \to \infty\)?
Logarithms \(\ln(n)\)
Polynomials \(n^k\)
Exponentials \(e^n\)
Factorials \(n!\)
Mission: Impossible?
Evaluate the following limit using L'Hôpital's Rule:
\(\lim_{n \to \infty} \frac{\ln(n^2)}{e^n}\)
Step 1: Continuous Function
Step 2: Differentiate
Advanced Limit Lab Worksheet Advanced Limit Lab
Experimental Calculus & Growth Analysis
Researcher:
ID:
Fastest Growth
\(n^n \gg n! \gg e^n \gg n^k \gg \ln(n)\)
Experiment 01: Log vs Poly
Determine the limit of the sequence \(a_n = \frac{(\ln n)^2}{n}\). Show each step: transition to continuous function, application of L'Hôpital's Rule, and final limit.
Experiment 02: The Exponential Wall
Evaluate the limit of \(b_n = \frac{n^2 + 3n}{e^{2n}}\). Note: This may require more than one application of L'Hôpital's Rule.
The Growth War Prediction
Without using calculus, predict the limit of the following sequence based on growth rates. Then, provide a one-sentence justification.
\(c_n = \frac{5n^{100}}{1.01^n}\)
Limit Prediction:
\(d_n = \frac{\ln(n^{500})}{\sqrt{n}}\)
Limit Prediction:
Justification for \(c_n\):
Advanced Limit Lab // Terminal Velocity of Functions
Steady Paths Slides Steady Paths
Monotonicity and Boundedness
Existence without Evaluation
Monotonicity
The sequence always moves in the same direction. (Increasing or Decreasing)
Boundedness
The sequence never crosses a certain value \(M\). (Upper or Lower Bound)
"If you keep going up but can't go past the ceiling, you must be approaching it."
Monotone Convergence Theorem
Every bounded and monotonic sequence of real numbers is convergent .
Scenario A
Increasing + Bounded Above
Scenario B
Decreasing + Bounded Below
Case Study: Recursive Roots
Consider the sequence defined by:
\(a_1 = \sqrt{2}\)
It's hard to find the general formula, but we can prove it's increasing and bounded by 2!
The Power of the MCT
If we know it converges, we can set \(L = \lim a_{n+1} = \lim a_n\).
\(L = \sqrt{2 + L}\)
The Infinite Horizon
We have moved from looking at graphs to calculating with algebra, trapping with bounds, and finally proving existence through logic.
You are ready for Calculus.
Convergence Proofs Handout Convergence Proofs
The Monotone Convergence Theorem
Handout // MCT-05
The MCT Statement
"If a sequence \(\{a_n\}\) is bounded above and increasing, then it converges. Similarly, if a sequence is bounded below and decreasing, then it converges."
Part 1: The Monotone Test
Consider the sequence \(a_n = \frac{n}{n+1}\).
Algebraic Difference Test
Show that \(a_{n+1} - a_n > 0\):
\(\frac{n+1}{n+2} - \frac{n}{n+1} = \) ...
Ratio Test (for positive terms)
Show that \(\frac{a_{n+1}}{a_n} > 1\):
\(\frac{n+1}{n+2} \div \frac{n}{n+1} = \) ...
Conclusion: Is the sequence increasing or decreasing?
Part 2: Finding the Ceiling
Prove that the sequence \(a_n = \frac{n}{n+1}\) is bounded above by 1.
Part 3: The Convergence Argument
Synthesize your findings. Why must this sequence converge? Does it converge to its bound? Explain.
Remember: The MCT tells you that a limit exists, but it doesn't always tell you what the limit is. For rational sequences, we use algebra to find the value; for recursive ones, we use the limit equation \(L = f(L)\).
Convergence Proofs // Logical Necessity
Plot Twists Exit Ticket Plot Twists Exit Ticket
Final Challenge
1. Classify the following sequence as Convergent, Divergent, or Oscillating:
\(a_n = \frac{n! + 1}{n^n}\)
Conv.
Div.
Osc.
2. Which theorem would you use to prove the limit of \(\frac{\cos(n)}{n^2}\)?
3. Briefly state the two requirements for the Monotone Convergence Theorem.
Duplicate Copy
1. Classify the following sequence as Convergent, Divergent, or Oscillating:
\(a_n = \frac{n! + 1}{n^n}\)
Conv.
Div.
Osc.
2. Which theorem would you use to prove the limit of \(\frac{\cos(n)}{n^2}\)?
3. Briefly state the two requirements for the Monotone Convergence Theorem.
"Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding." — William Paul Thurston
