Lesson 19 — Intuitive limit of sequences

Intuitive concept

The central question

Given a sequence $(a_n)$, toward what value (if any) do the terms approach as $n \to \infty$?

When that value exists, we say the sequence converges and write $\lim_{n \to \infty} a_n = L$.

Intuitive definition

$\lim a_n = L$ means: the terms $a_n$ become arbitrarily close to $L$ when $n$ is sufficiently large.

"Arbitrarily" and "sufficiently" are precisely what gets formalized with $\eps$ and $N$ in Lesson 41: $\forall \eps > 0, \exists N : n \geq N \Rightarrow |a_n - L| < \eps$

Notable limits

Operations with limits

If $\lim a_n = A$ and $\lim b_n = B$ (both finite):

• $\lim(a_n + b_n) = A + B$
• $\lim(a_n \cdot b_n) = A \cdot B$
• $\lim(a_n / b_n) = A/B$ (if $B \neq 0$)
• $\lim c \cdot a_n = c A$ ($c$ constant)

Sequences that do NOT converge

• Diverge to $\pm \infty$: $a_n = n$, $a_n = 2^n$.
• Oscillate: $a_n = (-1)^n$ — alternates between 1 and $-1$, tends to nothing.