10.1 & 10.2

Sequences

  • Definition of Sequence: An ordered list of numbers represented as ( { an } ), where ( a1, a2, a3, \ldots ).

  • Limit of a Sequence:

    • A limit of a sequence exists if ( { a_n } ) approaches a number ( L ) as ( n ) increases.

    • This means that by taking sufficiently large ( n ), ( a_n ) can get arbitrarily close to ( L ):

    • ( \lim{n \to \infty} an = L )

    • If the sequence does not approach a single number as ( n ) increases, it is said to diverge.

  • Divergence:

    • A divergent sequence is characterized by the absence of a limit, meaning the points ( (n, a_n) ) do not approach any horizontal line.

    • Example of a divergent sequence is ( \, { (-1)^n } ) which oscillates between ( 1 ) and ( -1 ).

    • Example of divergence in terms of limits: ( { a_n } = ext{Cos}(n\pi) ) diverges because as ( n ) varies, the sequence oscillates between ( 1 ) and ( -1 ).

  • Graphical Representations:

    • The graph of a convergent sequence approaches a particular horizontal line, whereas a divergent sequence does not.

Infinite Series

  • Definition of Infinite Series: An infinite series is defined as the sum of the terms of a sequence, represented as ( \Sigma a_k ), which equals:

    • ( S = a1 + a2 + a3 + \ldots + \Sigma ak )

  • Sequence of Partial Sums: The sequence of partial sums is denoted as ( { S_n } ), defined by:

    • ( S1 = a1 )

    • ( S2 = a1 + a_2 )

    • ( Sn = a1 + a2 + \ldots + an )

  • Convergence of Series:

    • If the sequence of partial sums ( S_n ) converges to a limit ( L ), then:

    • The infinite series converges to that limit.

    • Mathematically represented as:

      • ( \Sigma ak = \lim{n \to \infty} S_n = L )

      • If the sequence of partial sums diverges, then the infinite series also diverges.

Properties of Sequences

  • Bounded Monotonic Sequence: A sequence that is both bounded and monotonic (either non-decreasing or non-increasing) will converge.

  • Example:

    • Consider the sequence defined by:\

    • a) ( a1 = 2, a2 = 4, a_3 = 2 ) which diverges (oscillates).

    • b) Consider the sequence ( \, { (-0.5)^n } ), which converges towards 0, approaching it as n increases.

  • Behavior at Infinity:

    • Sequences that grow increasingly (e.g., ( \, {-3.2, 10.24, …} ) diverge as they tend to ( +\infty ) or oscillate around values without converging towards a single limit.

Limit Evaluations

  • Limit Evaluation Rule:

    • To determine convergence or divergence, consider sequences involving factorials, polynomial terms, and exponential terms as we evaluate their limits.

    • Example Evaluations:

    • ( \lim_{n \to \infty} \frac{(n + 1)!}{(n + 4)!} = 0 )

    • The factorials simplify as:

      • ( \lim{n \to \infty} \frac{(n + 1) n!}{(n + 4)(n + 3)(n + 2)(n + 1)n!} = \lim{n \to \infty} \frac{1}{(n + 4)(n + 3)(n + 2)} = 0 )

Test for Convergence or Divergence

  • Determining Convergence: Utilize tests like the ratio test for factorial sequences or the comparison test with known convergent series (e.g., geometric series).

  • Final Notes on Limits:

    • The process of evaluating limits, particularly regarding factorial patterns requires careful analysis to ensure the terms behave properly at infinity, ensuring we can confidently declare their convergent or divergent behavior.

  • For example, sequences defined as ( an = n^2 ) imply divergence since ( \lim{n \to \infty} a_n = \infty ).

Bounded Monotonic Sequence

A sequence is bounded monotonic if it satisfies two conditions:

  1. It is Bounded: There must be a real number ((M)) (an upper bound) such that every term in the sequence, $a_n$, is less than or equal to M(an≤M)M\left(a_{n}\le M\right) AND there must be a real number $N$ (a lower bound) such that every term in the sequence is greater than or equal to $N$ ($a_n \ge N$).

    • Essentially, the terms of the sequence are confined to a finite interval $[N, M]$.

  2. It is Monotonic: The terms of the sequence must be consistently increasing or consistently decreasing.

    • Monotonically Increasing: $a_n \le a_{n+1}$ for all $n$.

    • Monotonically Decreasing: $a_n \ge a_{n+1}$ for all $n$.

The Monotone Convergence Theorem

The significance of a sequence being both bounded and monotonic is guaranteed by the Monotone Convergence Theorem (MCT).

The theorem states that every bounded monotonic sequence must converge to a finite limit.

  • If a sequence is monotonically increasing and bounded above, it converges.1

  • If a sequence is monotonically decreasing and bounded below, it converges.2

This theorem is fundamental in calculus because it provides a way to prove that a limit exists without having to find the exact value of the limit.