Sequences and Convergence

A sequence is an ordered list of terms $a_n$. It converges when the terms approach one finite limit $L$ as $n \to \infty$ — and limit laws, the Squeeze Theorem, and monotone-bounded convergence are your three tools for pinning $L$ down.

By the end you'll be able to compute sequence limits with the limit laws, squeeze oscillating sequences like $\sin(n)/n$ to their limit, and prove a recursively defined sequence converges before you ever compute its limit.

Predict: as $n$ grows, do the two dashed squeeze bounds $\pm 1/n$ pinch the dots of $\sin(n)/n$ toward one value — even though $\sin(n)$ itself never settles? Slide n to check.

Each dot is one term $a_n$, plotted against its index $n$. For $\sin(n)/n$ the dashed bounds are the squeeze sandwich $-1/n \le a_n \le 1/n$; the horizontal line is the limit $L$. Switch to $(3n+1)/(n+2)$ to see a sequence that needs no squeezing — plain limit laws already give its limit. Hover any dot for exact values.

Sequence

Term index n = 1

a₁ = 0.841 · squeeze bounds: -1.000 ≤ a₁ ≤ 1.000 · limit L = 0

aₙ = sin(n)/n — terms, squeeze bounds ±1/n, and the limit line

terms aₙ current term bounds ±1/n limit L

A sequence $\{a_n\}$ converges to $L$ if its terms get arbitrarily close to $L$ for large $n$; limit laws handle algebraic combinations, the Squeeze Theorem tames oscillation, and monotone-bounded convergence proves a limit exists without computing it.

Visual

Picture the dots you just explored: a convergent sequence's dots crowd into an ever-thinner horizontal band around $L$ as you scan right. Divergence looks like dots marching off the top of the chart ($a_n \to \infty$) or bouncing forever without settling — $a_n = (-1)^n$ hops between $-1$ and $1$ and never picks a value. The Squeeze Theorem is literally visible: two dashed curves closing like jaws, with the sequence trapped between them. Wherever the jaws meet, the sequence must go.

Formal

$\lim_{n\to\infty} a_n = L$ means $a_n$ can be made arbitrarily close to $L$ by taking $n$ large enough. Limits obey the usual algebra: if $a_n \to A$ and $b_n \to B$, then $a_n \pm b_n \to A \pm B$, $a_n b_n \to AB$, and $a_n/b_n \to A/B$ (for $B \neq 0$). Since a sequence is a function sampled at integers, you may replace $n$ with a continuous variable $x$ and use calculus-1 tools like L'Hôpital's rule — but only after the swap, because L'Hôpital differentiates and you can't differentiate over the integers. Two more workhorses: Squeeze Theorem — if $a_n \le c_n \le b_n$ eventually and $\lim a_n = \lim b_n = L$, then $\lim c_n = L$; and the Monotone Convergence Theorem — every monotone bounded sequence converges. A useful special case: the geometric sequence $\{r^n\}$ converges exactly when $-1 < r \le 1$ — to $0$ if $|r| < 1$, to $1$ if $r = 1$, and diverges otherwise.

Worked example — the Squeeze Theorem

Find $\displaystyle\lim_{n\to\infty} \frac{\sin(n)}{n}$. The factor $\sin(n)$ oscillates forever, so limit laws stall — but it never leaves $[-1, 1]$. Since $-1 \le \sin(n) \le 1$, dividing by $n > 0$ gives $$-\frac{1}{n} \le \frac{\sin(n)}{n} \le \frac{1}{n}.$$ Both outer sequences $\to 0$, so by the Squeeze Theorem $\displaystyle\lim_{n\to\infty} \frac{\sin(n)}{n} = $ 0 — exactly the pinch you watched in the interactive above.

Your turn — monotone bounded convergence

Let $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{2 + a_n}$. There's no closed form for $a_n$, so first prove a limit exists: the sequence is increasing and bounded above by $2$, so the Monotone Convergence Theorem guarantees some limit $L$. Now take the limit of both sides of the recursion: $L = \sqrt{2 + L}$, so $L^2 = 2 + L$, giving $L^2 - L - 2 = 0$. Factor it: $(L - 2)(L + \_\_) = 0$, and since every term is positive, $L = \_\_$.

Reveal the answer

$L^2 - L - 2 = (L-2)(L+1) = 0$, so $L = 2$ or $L = -1$. The terms are all positive (they start at $\sqrt2$ and increase), so the limit is $L = 2$. Notice the two-step structure: the Monotone Convergence Theorem earned you the right to write "$L$" at all; only then did algebra find its value.

More info — why prove convergence before computing the limit?

In the recursive example, the step "take the limit of both sides" silently assumed the limit exists. That assumption can lie: apply the same trick to $a_1 = 1$, $a_{n+1} = 2a_n$ and you'd "solve" $L = 2L$ to get $L = 0$ — but the sequence $1, 2, 4, 8, \dots$ actually blows up. The Monotone Convergence Theorem is the safety check that makes the algebra legitimate: increasing + bounded above (or decreasing + bounded below) means a limit really is there before you solve for it. Beware the converse trap, too — bounded alone is not enough, as $(-1)^n$ shows. For more practice with this two-step pattern, work the examples in the Paul's Online Math Notes link under Dive deeper.

Check your understanding

Question 1 of 4

Find $\displaystyle\lim_{n\to\infty} \frac{3n^2+5}{n^2+2}$.

Question 2 of 4

$a_n = (-1)^n$ is bounded between $-1$ and $1$. What can you conclude about its convergence?

Question 3 of 4

Find $\displaystyle\lim_{n\to\infty} \frac{\cos(n)}{n^2}$.

Question 4 of 4

Find $\displaystyle\lim_{n\to\infty} \frac{2n + \cos(n)}{n}$. (Think about which tool handles each piece.)

Recap

A sequence $\{a_n\}$ converges to $L$ if $a_n$ gets arbitrarily close to $L$ as $n \to \infty$; otherwise it diverges (blows up or oscillates forever).
Limit laws mirror function limits: sums, products, and quotients of convergent sequences converge to the sums, products, and quotients of their limits. Swap $n$ for a continuous $x$ to borrow L'Hôpital's rule.
Geometric: $\{r^n\}$ converges exactly when $-1 < r \le 1$ — to 0 for $|r| < 1$, to 1 for $r = 1$.
Squeeze Theorem: if $a_n \le c_n \le b_n$ eventually and both bounds converge to the same $L$, then $c_n \to L$ — the tool for oscillating factors like $\sin(n)$ or $(-1)^n$.
Monotone Convergence Theorem: monotone + bounded ⇒ convergent — proves a limit exists (essential for recursive sequences) without computing it. Bounded alone is not enough.

Dive deeper

Paul's Online Math Notes — Sequences Work through convergence, limit laws, and the Squeeze Theorem for sequences.

Sources

Sequences, Limits, and Convergence