Real Number Sequences

Chapter 1: Real Sequences

• A real sequence can be seen as an infinite list of real numbers written in a certain order: a₁, a₂, a₃, a₄,...
• a₁ is the first term, a₂ is the second term, and so on
• The general term is denoted as aₙ
• The natural number n is called the index (or rank) of aₙ

Definition 1.1: Real Number Sequence

• A sequence of real numbers is a function a: N* → R, where n maps to a(n) = aₙ
• The sequence is written as (aₙ)ₙ∈N*, or simply (aₙ)

Example 1.2: Examples of Sequences

• aₙ = n corresponds to 1, 2, 3, 4, 5, ...
• aₙ = 1/n corresponds to 1, ½, ⅓, ¼, ⅕, ...
• aₙ = 42 corresponds to 42, 42, 42, 42, 42, ...
• aₙ = (-1)ⁿ corresponds to -1, 1, -1, 1, -1, 1, ...
• aₙ = n-th digit of the decimal expansion of π (π = 3.14159...) corresponds to 1, 4, 1, 5, 9, 2, ...

Ways to Define Sequences

• By an explicit formula of aₙ as a function of n, for example, aₙ = n²
• In a descriptive or implicit way, for example, aₙ = n-th digit of the decimal expansion of π
• By a recurrence relation, where aₙ is expressed in terms of the preceding terms, specifying some initial terms, for example, aₙ₊₁ = 2aₙ - 1, a₁ = 1

Graphical Representation of Sequences

• Sequences can be represented graphically on the real line R

1.2 Bounded and Monotonic Sequences

• The graph of the function a: N* → R, a(n) = aₙ can be plotted in the plane. The plotted points are of the form (n, aₙ)

1.2 Definitions 1.3: Bounded Sequences

• A sequence (aₙ) is bounded above if there exists M ∈ R such that aₙ ≤ M for all n ∈ N*
• Such an M is called an upper bound of (aₙ)
• A sequence (aₙ) is bounded below if there exists m ∈ R such that aₙ ≥ m for all n ∈ N*
• Such an m is called a lower bound of (aₙ)
• A sequence (aₙ) is bounded if it is both bounded above and bounded below

Remarks

• (aₙ) is bounded if and only if there exists C > 0 such that |aₙ| < C for all n ∈ N*
• A bounded above sequence (aₙ) has several upper bounds
• If M is an upper bound, then any N > M is also an upper bound
• The same applies to lower bounds

Examples 1.4: Examples of Bounded Sequences

• The sequence aₙ = 1/n is bounded below because aₙ ≥ 0 for all n, and bounded above because aₙ ≤ 1 for all n
• The sequence aₙ = n² is bounded below because aₙ ≥ 0 for all n
• However, it is not bounded above since there is no M such that n² ≤ M for all n
• Proof: If M ∈ R were an upper bound, then n² ≤ M ∀n, but taking n > √M, we have n² > |M| > M
• The sequence aₙ = -n is bounded above because aₙ ≤ 0 for all n
• However, not bounded below since there is no m such that -n ≥ m for all n
• The sequence aₙ = (-1)ⁿ is bounded because |aₙ| = 1, and thus -1 ≤ aₙ ≤ 1 for all n

1.2.1 Monotonicity Definitions 1.6: Monotonic Sequences

• (aₙ) is increasing if aₙ ≤ aₙ₊₁ for all n ∈ N*
• (aₙ) is strictly increasing if aₙ < aₙ₊₁ for all n ∈ N*
• (aₙ) is decreasing if aₙ ≥ aₙ₊₁ for all n ∈ N*
• (aₙ) is strictly decreasing if aₙ > aₙ₊₁ for all n ∈ N*
• (aₙ) is monotonic if it is either increasing or decreasing
• (aₙ) is strictly monotone if it is either strictly increasing or strictly decreasing

Examples 1.7: Examples of Monotonic Sequences

• aₙ = 1/n is strictly decreasing because an = 1/n > 1/(n+1) = aₙ₊₁ for all n
• aₙ = n² is strictly increasing because aₙ = n² < (n+1)² = aₙ₊₁
• aₙ = √n is increasing
• aₙ = (-1)ⁿ is neither increasing nor decreasing
• aₙ = ((-1)ⁿ)/(n+1) is neither increasing nor decreasing: a₂ > a₁ but a₃ < a₂
• an = (3n+4)/(n+1) is strictly decreasing

1.3 Sequences Tending to Infinity

• A sequence tends to infinity if, for any number M, the terms of the sequence become larger than M from a certain index onwards

Definition 1.8: Sequences Tending to Infinity

1. A sequence (aₙ) tends to +∞ if for every M > 0, there exists N₀ ∈ N* such that aₙ ≥ M for all n ≥ N₀
   • Denoted as limₙ→∞ aₙ = +∞, or aₙ → +∞
2. A sequence (aₙ) tends to -∞ if for every m < 0, there exists N₀ ∈ N* such that aₙ ≤ m for all n ≥ N₀
   • Denoted as limₙ→∞ aₙ = -∞, or aₙ → -∞

• A sequence that tends to infinity is said to diverge to infinity

Definition 1.9

• A sequence (aₙ) tends to -∞ if for every m < 0, there exists N₀ ∈ N* such that aₙ ≤ m for all n ≥ N₀
• The value M should be thought of as a "threshold"
• A sequence tending to +∞ will eventually exceed and remain above any threshold
• The index N₀ from which it exceeds the threshold depends on the value of M

Examples 1.10: Examples of Neighborhoods of Infinity

• {n ∈ N : n ≥ 173} is a neighborhood of infinity
• {n ∈ N : (n - 7)² > 4} contains a neighborhood of infinity
• {2n : n ∈ N} does not contain a neighborhood of infinity

1.4 Convergent Series

• A series (aₙ) converges toward a limit L

Definition 1.17

• A series (aₙ) converges toward a limited L ∈ R if for all ε > 0, there exists N ∈ N* such that for all n ≥ N, |aₙ – L| ≤ ε
• Denoted as limₙ→∞ aₙ = L or aₙ → L

Examples 1.18

• an=1/n, to show that (aₙ) → 0, for all ε > 0 you must find N such that |an-0| ≤ ɛ for all N ≥ n, which means n> 1/ε

1.5 Limit Properties

• Properties of Limits
• If the series is convergent it has one limit
• Every convergent series is limited
• If the series is convergent to 0, the absolute value is true to the same property, and vise versa

1.6 Combined Limits and Indeterminate Forms

• This section discusses combined limits and indeterminate forms in the context of real sequences. It covers cases where sequences tend to infinity and how their sums and products behave, including examples that result in indeterminate forms such as ∞ - ∞ and ∞ × 0
• L'Hôpital's rule is mentioned when combined with these limits