Sequences · study.

A sequence is a list of numbers written in a definite order,

a_{1}, a_{2}, a_{3}, \dots, a_{n}, \dots

The number $a_{1}$ is the first term, $a_{2}$ the second, and $a_{n}$ the $n$ th or general term. Every term has a successor, so the list never ends. Because each positive integer $n$ determines exactly one term $a_{n}$ , a sequence is a function whose domain is the set of positive integers. The only reason we write $a_{n}$ instead of $f (n)$ is convention; nothing else about the function idea changes.

The sequence ${a_{1}, a_{2}, a_{3}, \dots}$ is denoted ${a_{n}}$ or ${a_{n}}_{n = 1}^{\infty}$ . A sequence can be given three ways: by a formula for the general term, by listing terms, or by a rule that has no simple closed form.

By formula. $a_{n} = \frac{n}{n + 1}$ generates $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots$
By a recurrence. The Fibonacci sequence ${f_{n}}$ is defined by $f_{1} = 1$ , $f_{2} = 1$ , and $f_{n} = f_{n - 1} + f_{n - 2}$ for $n \geq 3$ , giving $1, 1, 2, 3, 5, 8, 13, 21, \dots$ — each term is the sum of the two before it.
By description alone. Let $a_{n}$ be the $n$ th decimal digit of $e$ ; then ${a_{n}} = {7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, \dots}$ is a perfectly well-defined sequence with no generating formula.

The index need not start at $1$ . The sequence ${n - 3}_{n = 3}^{\infty}$ begins at $n = 3$ , and ${cos (nπ /6)}_{n = 0}^{\infty}$ begins at $n = 0$ .

The limit of a sequence

Because a sequence is a function on the integers, it can be pictured either by plotting its terms on a number line or by plotting the points $(n, a_{n})$ in the plane. The graph is a set of isolated points, one above each integer. For $a_{n} = n / (n + 1)$ the points climb toward the line $y = 1$ : the gap

1 - \frac{n}{n + 1} = \frac{1}{n + 1}

can be made as small as we like by taking $n$ large enough. We write $lim_{n \to \infty} n / (n + 1) = 1$ .

The terms of

a_{n} = n / (n + 1)

are isolated points that climb toward the line

y = L = 1

; past some index

N

every term lies inside the band of half-width

ε

about

L

The picture suggests the intuitive definition: $a_{n} \to L$ if the terms get and stay arbitrarily close to $L$ . The precise version copies the definition of a limit at infinity, with the single change that $n$ ranges over integers.

Geometrically, no matter how narrow the band $(L - ε, L + ε)$ is drawn, all terms from some index $N + 1$ onward must fall inside it. A smaller $ε$ usually forces a larger $N$ .

If a function of a real variable agrees with the sequence at the integers, its limit at infinity carries over.

The converse fails: a sequence can converge while the interpolating function oscillates. The theorem still handles most evaluations, since it lets us reuse everything known about functions. In particular, since $lim_{x \to \infty} x^{- r} = 0$ for $r > 0$ ,

n \to \infty lim \frac{1}{n ^{r}} = 0 (r > 0) .

If $a_{n}$ grows without bound we write $lim_{n \to \infty} a_{n} = \infty$ , meaning that for every $M > 0$ there is an $N$ with $a_{n} > M$ whenever $n > N$ . Such a sequence still diverges; it simply diverges to infinity.

Three archetypes. Left: terms settle to a limit

L

(convergent). Middle: terms climb past every bound (divergent to

\infty

). Right: terms jump between two values forever and approach nothing (bounded, divergent).

Limit Laws for sequences

The Limit Laws hold for sequences with the same statements and near-identical proofs. If ${a_{n}}$ and ${b_{n}}$ converge and $c$ is constant, then

n \to \infty lim (a_{n} \pm b_{n}) = n \to \infty lim a_{n} \pm n \to \infty lim b_{n}, n \to \infty lim (c a_{n}) = c n \to \infty lim a_{n},

n \to \infty lim (a_{n} b_{n}) = (n \to \infty lim a_{n}) (n \to \infty lim b_{n}), n \to \infty lim \frac{a _{n}}{b _{n}} = \frac{lim _{n \to \infty} a _{n}}{lim _{n \to \infty} b _{n}} if n \to \infty lim b_{n} \neq = 0.

The Squeeze Theorem also transfers.

A companion fact handles alternating signs: if $lim_{n \to \infty} ∣ a_{n} ∣ = 0$ , then $lim_{n \to \infty} a_{n} = 0$ , because $- ∣ a_{n} ∣ \leq a_{n} \leq ∣ a_{n} ∣$ squeezes $a_{n}$ to $0$ .

Three tactics cover most evaluations:

Divide by the dominant power. For $lim_{n \to \infty} n / (n + 1)$ , divide top and bottom by $n$ : $n / (n + 1) = 1/ (1 + 1/ n) \to 1$ .
Push a continuous function inside. If $a_{n} \to L$ and $f$ is continuous at $L$ , then $f (a_{n}) \to f (L)$ . Since sine is continuous at $0$ , $sin (π / n) \to sin 0 = 0$ .
Squeeze when the terms trap between two known limits.

l'Hospital's Rule applies to functions of a real variable, not to a sequence directly; route it through the interpolating function and transfer the result back.

The Squeeze Theorem is the only recourse when no real-variable function interpolates the sequence at all.

Powers, oscillation, and divergence

Two sequences recur often enough to record. The sequence ${r^{n}}$ behaves according to the size of $r$ .

The sequence

r^{n}

converges only for

- 1 < r \leq 1

; outside that range it grows without bound or oscillates, and

r = - 1

gives the divergent flip-flop

1, - 1, 1, - 1, \dots

For $r > 1$ the terms grow to $\infty$ ; for $r \leq - 1$ they oscillate or grow in magnitude without settling. The case $r = - 1$ gives ${(- 1)^{n}} = - 1, 1, - 1, 1, \dots$ , which visits both $- 1$ and $1$ infinitely often and therefore approaches no single number. Oscillation between distinct values is the most common way a bounded sequence diverges.

Monotonic and bounded sequences

Two structural properties let us prove convergence even when the limit is unknown.

To show $a_{n} = 3/ (n + 5)$ is decreasing, note the denominator grows, so $3/ (n + 5) > 3/ (n + 6) = a_{n + 1}$ . For $a_{n} = n / (n^{2} + 1)$ the comparison is less obvious, so differentiate $f (x) = x / (x^{2} + 1)$ : $f^{'} (x) = (1 - x^{2}) / (x^{2} + 1)^{2} < 0$ for $x > 1$ , so $f$ decreases and the sequence decreases too.

Neither property alone forces convergence. The bounded sequence ${(- 1)^{n}}$ diverges, and the monotonic sequence $a_{n} = n$ diverges to infinity. Together, however, they are decisive.

An increasing sequence bounded above by

M

: the terms are forced to crowd together and approach a limit

L \leq M

that they never exceed.

The proof rests on the Completeness Axiom of the real numbers: a nonempty set bounded above has a least upper bound. Suppose ${a_{n}}$ is increasing and bounded above, and let $L$ be the least upper bound of the set ${a_{n} : n \geq 1}$ . Given $ε > 0$ , the number $L - ε$ is not an upper bound (since $L$ is the least one), so $a_{N} > L - ε$ for some $N$ . Because the sequence increases, $a_{n} \geq a_{N} > L - ε$ for all $n > N$ ; and $a_{n} \leq L$ always. Hence $0 \leq L - a_{n} < ε$ for $n > N$ — the definition of $lim_{n \to \infty} a_{n} = L$ . A symmetric argument, using the greatest lower bound, handles decreasing sequences.

The theorem certifies that a limit exists without producing its value. It underlies the convergence arguments for series of positive terms, whose partial sums increase.

Recursively defined sequences

The Monotonic Sequence Theorem is most useful on sequences given by a recurrence, where no formula for $a_{n}$ is available. Consider

a_{1} = 2, a_{n + 1} = \frac{1}{2} (a_{n} + 6) .

Computing terms gives $2, 4, 5, 5.5, 5.75, 5.875, \dots$ , suggesting an increasing sequence approaching $6$ .

Increasing. By induction: $a_{2} = 4 > 2 = a_{1}$ . If $a_{k + 1} > a_{k}$ , then $a_{k + 1} + 6 > a_{k} + 6$ , so $\frac{1}{2} (a_{k + 1} + 6) > \frac{1}{2} (a_{k} + 6)$ , i.e. $a_{k + 2} > a_{k + 1}$ . Thus $a_{n + 1} > a_{n}$ for all $n$ .

Bounded above by $6$ . Again by induction: $a_{1} = 2 < 6$ . If $a_{k} < 6$ , then $a_{k} + 6 < 12$ , so $a_{k + 1} = \frac{1}{2} (a_{k} + 6) < 6$ .

Being increasing and bounded, the sequence converges to some $L$ . To find $L$ , take the limit of both sides of the recurrence. Since $a_{n + 1} \to L$ and $a_{n} \to L$ ,

L = \frac{1}{2} (L + 6) ⟹ 2 L = L + 6 ⟹ L = 6.

This is the standard two-step method: prove convergence first, then solve a fixed-point equation. Solving $L = \frac{1}{2} (L + 6)$ is legitimate only once the Monotonic Sequence Theorem guarantees $L$ exists; otherwise the algebra can produce a false answer for a divergent sequence.

Cobweb view of

a_{n + 1} = \frac{1}{2} (a_{n} + 6)

: iterating bounces between the map

y = \frac{1}{2} (x + 6)

and the diagonal

y = x

, spiralling into the fixed point where the two lines cross.

A second recurrence shows the method when the map is a square root rather than a line.

The recursion

a_{n + 1} = 2 + a_{n}

starting from

2

: terms increase and stay below the dashed line

L = 2

, crowding toward it.

Summary of sequence facts

Idea	Statement
Limit of a sequence	$a_{n} \to L$ : terms stay within $ε$ of $L$ past some $N$
Powers $r^{n}$	converge iff $- 1 < r \leq 1$ (to $0$ , or to $1$ at $r = 1$ )
Squeeze Theorem	$a_{n} \leq b_{n} \leq c_{n}$ with $a_{n}, c_{n} \to L$ forces $b_{n} \to L$
Monotonic Sequence Theorem	bounded $+$ monotonic $\Rightarrow$ convergent
Two-step recursion method	prove convergence, then solve the fixed-point equation $L = g (L)$

The sum of an infinite series is defined as the limit of its sequence of partial sums, so every result above transfers directly to series.