Sequences: Convergence and Divergence
Define a sequence as a function on positive integers. Determine convergence or divergence of sequences algebraically and using the Squeeze Theorem. Apply L'Hôpital's Rule to limits of the form aₙ = f(n).
A lesson of AP Calculus BC, in Infinite Sequences and Series, about 10 minutes.
Sequences
By the end of this lesson, you will be able to determine convergence or divergence of sequences and find their limits.
Sequences
A sequence ${a_n}$ is an ordered list of numbers: $a_1, a_2, a_3, \ldots$
Convergence: $\lim_{n\to\infty} a_n = L$ (exists and is finite) → sequence converges to $L$.
Divergence: the limit doesn't exist or is $\pm\infty$.
Examples
| Sequence | Formula | Limit | Convergence |
|---|---|---|---|
| $1, 1/2, 1/3, 1/4, \ldots$ | $1/n$ | 0 | Converges |
| $1, 2, 3, 4, \ldots$ | $n$ | $\infty$ | Diverges |
| $-1, 1, -1, 1, \ldots$ | $(-1)^n$ | DNE | Diverges |
| $1, 3/4, 5/9, 7/16, \ldots$ | $(2n-1)/n^2$ | 0 | Converges |
Finding Limits of Sequences
Treat as a continuous function and find the limit of $f(x)$ as $x\to\infty$.
$\lim_{n\to\infty}\frac{n^2+3n}{2n^2-1} = \lim_{x\to\infty}\frac{x^2+3x}{2x^2-1} = \frac{1}{2}$ (ratio of leading coefficients)
Useful limits:
- $r^n \to 0$ if $|r| < 1$; diverges if $|r| > 1$
- $\frac{n!}{n^n} \to 0$
- $n^{1/n} \to 1$
- $r^n/n! \to 0$ for any $r$ (factorial grows faster)
L'Hôpital's Rule can be applied to sequences by treating $n$ as continuous.
Monotone Sequences and Boundedness
A monotone bounded sequence converges (Monotone Convergence Theorem).
Monotone increasing: $a_{n+1} \geq a_n$ for all $n$. Bounded above: $a_n \leq M$ for all $n$.
If both hold → the sequence converges.
Summary
Sequence converges if $\lim_{n\to\infty}a_n = L$ (finite).
Technique: treat as a function of $x$; apply limit laws or L'Hôpital's.