Sequences: complete lesson (arithmetic, geometric, limits, induction)

What is a sequence?

A numerical sequence is a function $u : \mathbb{N} \to \mathbb{R}$ that maps an integer $n$ to a real number $u_n$.

Notation: $(u_n)_{n \in \mathbb{N}}$ or simply $(u_n)$.

Two ways to define a sequence

Explicit: $u_n = f(n)$. Example: $u_n = 2n + 1$.

Recursive: $u_0$ given and $u_{n+1} = f(u_n)$. Example: $u_0 = 1$, $u_{n+1} = 2 u_n + 1$.

Monotonicity

• $(u_n)$ increasing if $u_{n+1} \geq u_n$ for all $n$.
• $(u_n)$ decreasing if $u_{n+1} \leq u_n$.
• $(u_n)$ constant if $u_{n+1} = u_n$.

Method: study the sign of $u_{n+1} - u_n$, or (if $u_n > 0$) compare $\dfrac{u_{n+1}}{u_n}$ to $1$.

Arithmetic sequences

$u_{n+1} = u_n + r$ where $r$ is the common difference.

Explicit form: $u_n = u_0 + n \cdot r$ (or $u_n = u_p + (n-p) \cdot r$).

Sum: $\displaystyle\sum_{k=0}^{n} u_k = (n+1) \cdot \dfrac{u_0 + u_n}{2}$.

Geometric sequences

$u_{n+1} = q \cdot u_n$ where $q$ is the common ratio.

Explicit form: $u_n = u_0 \cdot q^n$ (or $u_n = u_p \cdot q^{n-p}$).

Sum ($q \neq 1$): $\displaystyle\sum_{k=0}^{n} u_k = u_0 \cdot \dfrac{1 - q^{n+1}}{1 - q}$.

Limit of a sequence

• $(u_n)$ converges to $\ell$ if $u_n$ gets arbitrarily close to $\ell$ as $n \to +\infty$. We write $\lim u_n = \ell$.
• $(u_n)$ diverges to $+\infty$ if $u_n$ becomes arbitrarily large.

Reference limits

Sequence	Limit
$1/n$	$0$
$q^n$ if $	q
$q^n$ if $q > 1$	$+\infty$
$q^n$ if $q \leq -1$	diverges, no limit
$\sqrt{n}$	$+\infty$

Convergence theorems

Monotone convergence: Any increasing and bounded above sequence converges. Any decreasing and bounded below sequence converges.

Squeeze theorem: If $v_n \leq u_n \leq w_n$ and $v_n, w_n \to \ell$, then $u_n \to \ell$.

Proof by induction

To show $P(n)$ true for all $n \geq n_0$:

1. Base case: check $P(n_0)$.
2. Inductive step: assume $P(n)$ true, show $P(n+1)$.
3. Conclusion: $P(n)$ is true for all $n \geq n_0$.

Common mistakes

1. Confusing explicit and recursive sequences.
2. Dividing by $u_n$ without checking $u_n \neq 0$ in geometric sequences.
3. Skipping the base case in induction.
4. Writing $\lim u_n \cdot v_n = \lim u_n \cdot \lim v_n$ without checking hypotheses ($\infty \times 0$ is indeterminate).