Lesson 16 — Numerical sequences

Definition and properties

How to describe a sequence

1. Explicit formula (general term): $a_n = 2n + 1$ — terms $3, 5, 7, 9, \ldots$
2. Recurrence: $a_1 = 1$, $a_{n+1} = a_n + 2$ — same result.
3. Description: "n-th prime number" — $2, 3, 5, 7, 11, 13, \ldots$ (no closed form).

Monotonicity

• Increasing: $a_{n+1} > a_n \quad \forall n$.
• Non-decreasing: $a_{n+1} \geq a_n$.
• Decreasing: $a_{n+1} < a_n$.
• Constant: $a_{n+1} = a_n$.

Boundedness

$(a_n)$ is bounded if there exists $M > 0$ with $|a_n| \leq M$ for all $n$. Bounded above if $a_n \leq M_+$; below if $a_n \geq M_-$.

Intuitive convergence (formalized in Lesson 19)

$(a_n)$ converges to $L$ if "$a_n$ gets arbitrarily close to $L$ when $n$ is large". Formally (Lesson 41 — Trim 5): $\lim_{n \to \infty} a_n = L \iff \forall \eps > 0, \exists N : n \geq N \Rightarrow |a_n - L| < \eps$