Mathematics of Limits, Sequences, and Factorials

Limits and Convergence of Sequences

  • Basic Definition of Limits

    • If sequences $an$ and $bn$ converge, then the limit of their sum is the sum of their limits: lim⁡<em>n→∞(a</em>n+b<em>n)=lim⁡</em>n→∞a<em>n+lim⁡</em>n→∞bn\lim<em>{n \to \infty} (a</em>n + b<em>n) = \lim</em>{n \to \infty} a<em>n + \lim</em>{n \to \infty} b_n
    • Similar concept applies for products and quotients:
    • Product Rule: lim⁡<em>n→∞(c</em>n⋅d<em>n)=lim⁡</em>n→∞c<em>n⋅lim⁡</em>n→∞dn\lim<em>{n \to \infty} (c</em>n \cdot d<em>n) = \lim</em>{n \to \infty} c<em>n \cdot \lim</em>{n \to \infty} d_n
    • Quotient Rule: lim⁡<em>n→∞(c</em>nd<em>n)=lim⁡</em>n→∞c<em>nlim⁡</em>n→∞dn\lim<em>{n \to \infty} \left( \frac{c</em>n}{d<em>n} \right) = \frac{\lim</em>{n \to \infty} c<em>n}{\lim</em>{n \to \infty} d_n}

  • Sandwich Theorem (Squeeze Theorem)

    • If $an \leq bn \leq c_n$ for all $n$, and:
    • lim⁡<em>n→∞a</em>n=lim⁡<em>n→∞c</em>n=L\lim<em>{n \to \infty} a</em>n = \lim<em>{n \to \infty} c</em>n = L
    • Then lim⁡<em>n→∞b</em>n=L\lim<em>{n \to \infty} b</em>n = L
    • This theorem implies that if sequences $an$ and $cn$ converge to the same limit, so does the sequence $b_n$.

Factorials

  • Definition of Factorial
    • Notation:
    • $n! = n \cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1$
    • Example Calculation:
    • 4!=4⋅3⋅2⋅1=244! = 4 \cdot 3 \cdot 2 \cdot 1 = 24
    • 5!=5⋅4⋅3⋅2⋅1=1205! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120

Example Problem Involving Factorials

  • Consider the sequence: an=n!nna_n = \frac{n!}{n^n}
  • Limit Calculation
    • Breakdown of $n!$:
    • n!=n⋅(n−1)⋯3⋅2⋅1n! = n \cdot (n - 1) \cdots 3 \cdot 2 \cdot 1
    • Breakdown of $n^n$:
    • nn=n⋅n⋯nn^n = n \cdot n \cdots n (n terms)
  • Rewrite as:
    • an=n⋅(n−1)⋯3⋅2⋅1n⋅n⋯n=(n−1)n⋅(n−2)n⋯3n⋅2n⋅1na_n = \frac{n\cdot (n-1)\cdots 3 \cdot 2 \cdot 1}{n \cdot n \cdots n} = \frac{(n-1)}{n} \cdot \frac{(n-2)}{n} \cdots \frac{3}{n} \cdot \frac{2}{n} \cdot \frac{1}{n}
    • Analysis shows:
    • Each term approaches 0 as $n \to \infty$
  • Using the Sandwich Theorem
    • Conclude:
    • lim⁡<em>n→∞a</em>n=0\lim<em>{n \to \infty} a</em>n = 0

Absolute Values and Convergence

  • Example of sequence of absolute values approaches to zero:
    • Sequence: $\left| \frac{1}{n} \right|$ leads to:
    • Each term: 1,12,13,14,…1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots approaches 0
    • This sequence is therefore converging to 0.

Recursive Sequences

  • Definition: A sequence defined using recursion involves past terms for next term definition.
  • Example: Define $an$ recursively: - Given $a1 = 2$, find $a2, a3, \ldots$:
    • Recursive formula: a<em>n+1=a</em>n+62a<em>{n+1} = \frac{a</em>n + 6}{2}
  • Calculation Steps:
    • Calculate $a_2$:
    • a<em>2=a</em>1+62=2+62=4a<em>2 = \frac{a</em>1 + 6}{2} = \frac{2 + 6}{2} = 4
    • Calculate $a_3$:
    • a<em>3=a</em>2+62=4+62=5a<em>3 = \frac{a</em>2 + 6}{2} = \frac{4 + 6}{2} = 5
    • Proceed similarly for higher indices.

Convergence of Sequences and Limits

  • Boundedness:
    • Sequence is bounded above if there exists an upper limit for its values.
    • Lower bound likewise prevents approaches to negative infinity.
    • Example 'bounded from above':
    • 1,12,13,…1, \frac{1}{2}, \frac{1}{3}, \ldots
    • Upper bound is 1, while lower bound is 0.
  • Unbounded Sequence
    • Example of unbounded: $n^2$, where values increase indefinitely:
    • Sequence: 1, 4, 9, 16, 25, …
  • Monotonic Sequences
    • A sequence which is either non-increasing or non-decreasing.
    • Theorem: If a sequence is both bounded and monotonic, then it converges.

Homework and Class Administration

  • Assignment due date was discussed as possibly moved to Wednesday or Thursday but final agreement was to keep it Monday.
  • Emphasis on completing before the deadline without stress.