Calc_II_10.1

Sequences

• Definition: A sequence is a list of numbers in a definite order.

• Terms in a sequence:

  • a: The first term.

  • a₂: The second term.

  • aₙ: The n-th term (also denoted as G(aₙ) or Y(aₙ)).

• A sequence can be viewed as an ordered collection of numbers defined by a function F on a set of sequential integers:

  • Examples:

    • F(1) = a, F(2) = a₂, F(n) = aₙ.

• Examples of sequences:

  • a)

    • Sequence: 0, 1, 2, 5, 2, 55...

  • b)

    • Sequence: 1, -1, 1, -1, 1, -1...

• Note: A formula may not always exist to describe the n-th term of a sequence.

• Recursive Formula example:

  • Fibonacci Sequence:

    • a₁ = 1, a₂ = 1, aₙ = aₙ₋₁ + aₙ₋₂.

    • Resulting Sequence: 1, 1, 2, 3, 5, 8, 13, 21...

Convergence and Divergence

• Question: Do the numbers in a sequence approach a single value as n increases?

• Evaluate:

  • Lim aₙ as n approaches ∞.

  • If the limit is a number (L), we say the sequence converges to L.

  • If the limit does not exist (DNE), we say the sequence diverges.

• If a function f(x) = L and f(n) = aₙ, where n is an integer, then:

  • Lim aₙ = L.

• Limit properties for sequences:

  1. Lim(a + b) = Lim a + Lim b.

  2. Lim(ca) = cLim a (where c is a constant).

  3. Lim(aₙbₙ) = Lim aₙ * Lim bₙ.

  4. If bₙ approaches 0, the limits can behave differently; apply L'Hôpital's Rule or Squeeze Theorem accordingly.

Examples of Convergence/Divergence

• Determining convergence/divergence:

  • a) Lim (n+1) converges to a constant.

  • b) Lim (n-3) diverges to ∞.

  • c) Sequence alternating between 1 and -1 diverges (oscillates, DNE).

  • d) For sequence (n) converging to the limit (L), we cannot apply L'Hôpital's Rule to sequences, refer to the corresponding function instead for natural domain analysis.

  • e) For aₙ = 2^(3 + 3n - 1), this diverges as n approaches ∞.

  • f) Sequence involving sin(n) converges to 0 using Squeeze Theorem.

Additional Notes

• If aₙ = 0, then the limit is 0.

• Examples:

  • For the limit of (-1)ⁿ as n approaches ∞, if terms approach c, analyze convergence.

• Determine if sequences converge/diverge based on behavior and formula interpretations.

• Example: 2^n diverges as n approaches ∞ (goes to ∞).

• Understanding convergence is essential for sequences in higher math.