Summary of 10.1 Sequences

Main Ideas and Concepts

• Definition of a Sequence:

  A sequence is defined as a list of numbers arranged in a specific order, often denoted by terms such as \( a_1, a_2, a_3, \ldots \).

• Types of Sequences:
  • Infinite Sequences: Sequences that continue indefinitely.
  • Sequences can be represented by formulas, allowing for the generation of terms based on an index.
• Convergence and Divergence:

  A sequence converges if it approaches a specific value as \( n \) increases; it diverges if it does not approach any limit or approaches infinity.

  Example: The sequence \( 1/n \) converges to 0 as \( n \) approaches infinity, while the sequence \( 1, -1, 1, -1, \ldots \) diverges.

• Limit Theorems:

  Limits can be calculated for sequences, and certain rules apply, such as:

  • The limit of a sum is the sum of the limits.
  • The limit of a product is the product of the limits.

  Special cases include using L'Hôpital's rule for indeterminate forms.

• Common Limits:

  Several common limits are identified, such as \( \ln(n)/n \) converging to 0 and \( (1 + x/n)^n \) converging to \( e^x \).

• Recursive Definitions:

  Some sequences can be defined recursively, where each term is derived from previous terms, like the Fibonacci sequence.

• Bounded and Monotonic Sequences:

  A sequence is bounded if it has upper and/or lower limits.

  Monotonic sequences are either non-decreasing or non-increasing.

  The Monotonic Sequence Theorem states that if a sequence is both bounded and monotonic, it converges.

Methodology and Instructions

• Calculating Limits:
  • To find the limit of a sequence as \( n \) approaches infinity:
  1. Identify the sequence and express it in a suitable form.
  2. Apply limit rules (sum, product, etc.) as necessary.
  3. Use L'Hôpital's rule for indeterminate forms.
  4. Check for Convergence or Divergence.
• Identifying Monotonicity:
  • A sequence is non-decreasing if \( a_n \leq a_{n+1} \).
  • A sequence is non-increasing if \( a_n \geq a_{n+1} \>.

Speakers or Sources

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