Lecture on Sequences (Section 11.1)

Introduction to Sequences

• Definition: A sequence is a discrete function where the domain is a set of positive integers (e.g., 1, 2, 3, 4, ...).
• Notation: Sequences are often denoted as ( a_n ).
• Example: ( a_n = \frac{1 + n}{2^n} ).

Key Questions

• Is the sequence increasing or decreasing?
• Is the sequence bounded or unbounded?

Example Sequences

Example 1

• For ( a_n = \frac{n}{n+1} ):
  • Calculate terms: ( a_1 = \frac{1}{2}, a_2 = \frac{2}{3}, a_3 = \frac{3}{4} ).
  • The sequence approaches 1 as ( n \rightarrow \infty ).
  • Limit: ( \lim_{{n \to \infty}} \frac{n}{n+1} = 1 ).
  • Bounded: Between ( \frac{1}{2} ) (lower bound) and 1 (upper bound).

Example 2

• For an alternating sequence ( a_n = (-1)^n \frac{n+1}{3^n} ):
  • Alternating series changes signs.
  • Values decrease in magnitude, converging to 0.
  • Limit: 0.

Recursive Sequences

• Example: Fibonacci sequence defined recursively.
  • Starts with 1, 1: ( F_1 = 1, F_2 = 1 )
  • ( F_3 = F_2 + F_1 = 2 ).
  • Continues with ( F_4 = 3, F_5 = 5, F_6 = 8 ).
  • Limit: Diverges to infinity.

Convergence and Divergence

• Convergent Sequences: Sequence approaches a limit L.
  • Example: ( a_n = \frac{1 + n}{2^n} ) converges to 0.
• Divergent Sequences: Sequence limit is ( \infty ) or (-\infty ).
  • Example: ( a_n = \frac{2^n}{1+n} ) diverges to infinity.

Limit Properties

• Sum, difference, product, and quotient of limits follow standard limit laws if individual limits exist.
• Example: Limit of the sum of sequences is the sum of the limits.

Squeeze Theorem

• If ( a_n \leq b_n \leq c_n ) and ( \lim a_n = \lim c_n = L ), then ( \lim b_n = L ).

Monotonic Sequences

• Monotonic Increasing: Always increasing (e.g., ( a_n = \frac{n}{n+1} )).
• Monotonic Decreasing: Always decreasing (e.g., ( a_n = \frac{1}{n} )).

Bounded Sequences

• Sequences are bounded if confined within an upper and lower limit.

Theorem: Bounded and Monotonic

• Every bounded and monotonic sequence is convergent.

Assignments

• Complete exercises #23, #27, #73, #75, #79 from the textbook.
• Watch videos provided on Canvas for solutions and further explanations.