What does it mean for a sequence to have a limit?

That its terms come arbitrarily close to one number g and stay there. Precisely: for every epsilon greater than zero, however small, there is an index N such that every term beyond N differs from g by less than epsilon. Such a sequence is called convergent and g is its limit.

How does a convergent sequence differ from a divergent one?

A convergent one has a limit that is a number. A divergent one has none — it either grows without bound, like aₙ = n², or falls without bound, or heads nowhere at all, like aₙ = (−1)ⁿ jumping between −1 and 1. Writing that the limit is infinity describes the manner of divergence, not the existence of a limit.

Do the terms have to reach the limit?

No, and usually they do not. The sequence aₙ = 1/n has limit 0, yet no term of it is zero. A limit is the number the terms approach, not a number they take.

How do I compute the limit of a quotient of two polynomials?

Divide the numerator and the denominator by the highest power of n in the denominator, then use the fact that every term with n in the denominator tends to zero. The outcome depends on the degrees: equal degrees give the ratio of the leading coefficients, a lower-degree numerator gives zero, and a higher-degree numerator makes the sequence divergent.

Where does the number e come from?

It is the limit of the sequence (1 + 1/n) to the power of n, so a limit is what defines it — no finite formula does. The successive terms are 2, 2.25, 2.37, 2.44 and so on, converging to 2.718281828… The number e is the base of the natural logarithm and describes continuous compounding, that is interest credited at every instant.

When does a geometric series have a sum?

Exactly when the absolute value of the ratio is smaller than 1. Then q to the power of n tends to zero, the partial sums approach a₁/(1 − q), and that number is called the sum of the series. For |q| greater than or equal to 1 the partial sums converge to nothing and there is no sum.

How do I turn a repeating decimal into a fraction?

Treat the expansion as a geometric series. The notation 0.(3) means 0.3 + 0.03 + 0.003 + …, a series with a₁ = 0.3 and q = 0.1, so its sum is 0.3/(1 − 0.1) = 3/9 = 1/3. The same route gives 0.(12) = 12/99 = 4/33, and in short: the repeating block on top, as many nines below as the block has digits.

Is 0.(9) really equal to 1?

Yes, and it is not an approximation. The series 0.9 + 0.09 + 0.009 + … has a₁ = 0.9 and q = 0.1, so its sum is 0.9/(1 − 0.1) = 0.9/0.9 = 1. The notations 0.(9) and 1 are two names for the same number, just as 1/2 and 0.5 are.

Advanced level

Limit of a sequence and the geometric series

Infinitely many terms, a finite total — provided the ratio is smaller than one in absolute value. See what it means precisely for a sequence to approach a limit, how limits of polynomial quotients are computed, where the number e comes from, and why 0.(3) really is one third.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Number sequencesA sequence is a function that assigns a number to every natural number — which is exactly why its graph is a set of separate dots rather than a curve. See how a general formula works, how a recursive one differs from it, and how a single subtraction settles whether a sequence increases.
Arithmetic and geometric sequencesTwo sequences describe almost everything that grows regularly: one keeps adding the same amount, the other keeps multiplying by the same amount. Meet the formulas for the nth term and for the sum of the first n terms of both, the mean property, and the reason a savings account is a geometric sequence.

Where this is used

Real situations where you count exactly the way this lesson teaches:

A bouncing ball
A ball dropped from 2 m returns to 60% of the previous height after every bounce. The bounce heights form a geometric sequence with q = 0.6, and the whole distance travelled is 2 + 2 · (1.2 + 0.72 + 0.432 + …) = 2 + 2 · 1.2/(1 − 0.6) = 8 metres. There are infinitely many bounces and the distance is finite — which is exactly the situation a convergent series describes.
The multiplier effect
A government spends 1 billion on an investment. Those who receive the money spend 80% of it onward, the next recipients spend 80% again — a geometric series with q = 0.8. An economist sums it with 1/(1 − 0.8) = 5, so total demand rises by 5 billion, five times the original outlay. At a propensity to consume of 0.5 the multiplier would drop to 2.
Successive approximation
A calculator has no dedicated square-root circuit: it computes the root with the sequence aₙ₊₁ = (aₙ + 2/aₙ)/2, convergent to √2. Starting from a₁ = 2 we get 1.5, then 1.41666, then 1.4142157 — after four steps six decimal places already agree. A programmer stops the iteration once |aₙ₊₁ − aₙ| drops below a chosen epsilon, which is precisely what the definition of a limit says.
A fractal in graphics
The Koch snowflake is built by repeatedly replacing every edge with four shorter ones. The perimeter is multiplied by 4/3 each time, so it grows without bound — a divergent sequence. The area, however, is the sum of a convergent series and settles at 8/5 of the starting triangle, that is 1.6 times as much. A graphics engine can therefore render a figure with an infinite perimeter and a finite surface.

All formulas

Limit of a sequence
$\lim_{n \to \infty} a_n = g$
a sequence convergent to the number g
Definition of the limit
$\forall \varepsilon > 0 \;\; \exists N \;\; \forall n > N : \; |a_n - g| < \varepsilon$
from some point on, every term lies closer to g than epsilon
The basic limit
$\lim_{n \to \infty} \frac{1}{n} = 0$
every quotient limit is derived from it
Quotient of polynomials
$\lim_{n \to \infty} \frac{a n^k + \ldots}{b n^k + \ldots} = \frac{a}{b}$
for equal degrees — the ratio of the leading coefficients
The number e
$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828$
a limit that is the definition of a constant
Sum of a geometric series
$S = \frac{a_1}{1 - q}, \qquad |q| < 1$
only for a convergent one — otherwise there is no sum

Infinitely many terms and a finite total — it sounds like a contradiction and is an ordinary calculation. All it needs is one idea: the limit.

Convergent and divergent sequences

Look at the sequence $a_n = 2 + \frac{1}{n}$ :

3, \quad 2.5, \quad 2.\overline{3}, \quad 2.25, \quad 2.2, \quad \ldots

The terms keep dropping, yet never fall below $2$ — and never reach that $2$ either. We still say they tend to $2$ .

The sequence a(n) = 2 + 1/n and its limit. The dots come down to the helper line without touching it — a limit is what the terms approach, not what they take.

The definition

The phrase about coming arbitrarily close has to be written so that it can be checked. That is what the definition does:

\lim_{n \to \infty} a_n = g \quad \Longleftrightarrow \quad \forall \varepsilon > 0 \;\; \exists N \;\; \forall n > N : \; |a_n - g| < \varepsilon

In plain words: pick any radius $\varepsilon$ around the number $g$ , however small; from some index $N$ on, every further term already lies inside it. At most finitely many opening terms may stay outside.

Let us check that on our sequence. For $\varepsilon = 0.01$ we look for an $N$ beyond which $|a_n - 2| = \frac{1}{n} < 0.01$ :

\frac{1}{n} < 0.01 \quad \Longleftrightarrow \quad n > 100

So $N = 100$ suffices. For $\varepsilon = 0.0001$ it comes out as $N = 10\,000$ — and so on for every $\varepsilon$ , however tiny. That is precisely the proof that the limit is $2$ .

A sequence whose limit is a number is called convergent. Every other one is divergent, and divergence wears three different faces:

sequence	behaviour	notation
$a_n = 2 + \frac{1}{n}$	convergent to $2$	$\lim a_n = 2$
$a_n = n^2$	grows without bound	$\lim a_n = +\infty$
$a_n = (-1)^n$	jumps between $-1$ and $1$	the limit does not exist

The notation $\lim a_n = +\infty$ is a convenient shorthand describing how a sequence diverges. Infinity is not a number, so such a sequence is not convergent — the equals sign notwithstanding.

Limits of polynomial quotients

The most frequent calculation of this topic. The basic limit everything follows from is

\lim_{n \to \infty} \frac{1}{n} = 0

and with it $\frac{1}{n^2} \to 0$ , $\frac{1}{n^3} \to 0$ and so on. The whole procedure comes down to one move: divide the numerator and the denominator by the highest power of $n$ in the denominator.

Find the limit of the sequence aₙ = (3n² + 5n − 1)/(2n² + 7).

The outcome depends only on the two degrees:

equal degrees — the limit is the ratio of the leading coefficients;
numerator of lower degree — the limit is $0$ , because the denominator grows faster;
numerator of higher degree — the sequence diverges to $+\infty$ or $-\infty$ .

Find the limit of the sequence aₙ = (4n + 3)/(n² − 2).

The same arithmetic returns later for the limit of a function at infinity. The difference lies in the object, not in the calculation: there the argument runs through all the real numbers, here through the naturals only.

The number e

One limit deserves a place of its own, because it defines a constant:

e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n

It looks innocent: the base tends to $1$ and the exponent to infinity. Computed separately that would give $1$ to an infinite power — which is no result at all but an indeterminate form. The whole expression has to be handled at once:

$n$	$10$	$100$	$1000$	$100\,000$
$\left(1 + \frac{1}{n}\right)^n$	$2.5937$	$2.7048$	$2.7169$	$2.71827$

e \approx 2.718281828

The number $e$ is the base of the natural logarithm, and in finance it describes continuous compounding — interest credited not once a year but at every instant.

The geometric series

A series is the sum of all the terms of a sequence. Take the geometric sequence with $a_1 = 1$ and $q = \frac{1}{2}$ and add its terms one by one:

S_1 = 1, \quad S_2 = 1.5, \quad S_3 = 1.75, \quad S_4 = 1.875, \quad S_5 = 1.9375, \; \ldots

Partial sums of the series 1 + 1/2 + 1/4 + … Each step adds half of what is still missing from 2, so the sequence of sums converges.

The sequence of partial sums converges to $2$ — and that limit is what we call the sum of the series.

Where the formula comes from

Take the formula for the sum of the first $n$ terms and let $n$ go to infinity:

S_n = a_1 \cdot \frac{1 - q^n}{1 - q}

For $|q| < 1$ the power $q^n$ tends to zero — multiplying by a fraction smaller than one shrinks the number at every step. What remains is

S = \frac{a_1}{1 - q}, \qquad |q| < 1

The condition $|q| < 1$ is part of the formula, not a footnote. For $q = 2$ the partial sums $1, 3, 7, 15, \ldots$ grow without bound, while the formula would return $\frac{1}{1-2} = -1$ — a negative number as the total of positive terms. The formula may be applied only after the condition has been checked.

Find the sum of the geometric series with a₁ = 6 and q = 1/3.

A repeating decimal is a series

The expansion $0.(3)$ is not an almost anything — it is the sum of a series:

0.(3) = 0.3 + 0.03 + 0.003 + \ldots

That is a geometric series with $a_1 = 0.3$ and $q = 0.1$ , and since $|0.1| < 1$ it has a sum:

S = \frac{0.3}{1 - 0.1} = \frac{0.3}{0.9} = \frac{3}{9} = \frac{1}{3}

Every other repeating block works the same way — for $0.(12)$ the first term is $0.12$ and the ratio $0.01$ :

S = \frac{0.12}{1 - 0.01} = \frac{0.12}{0.99} = \frac{12}{99} = \frac{4}{33}

Hence the schoolroom shortcut: the repeating block on top, as many nines below as the block has digits. Turning a decimal into a fraction stops being a rule to memorise and becomes a calculation you can derive.

The same calculation settles the most famous argument in school mathematics:

0.(9) = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1

$0.(9)$ and $1$ are two notations for the same number — not two numbers lying very close together.

Exercises

The set drills two calculations: the limit of a quotient of polynomials (divide by the highest power) and the sum of a convergent geometric series (check $|q| < 1$ first, then apply the formula). Some answers are fractions — type those as 3/4; any equivalent spelling of the same number is accepted. Divergent sequences and infinite limits stay with the examples above, because their answer is a symbol rather than a number.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

lim(n → ∞) (n + 2)/(4n + 5) =

Common mistakes

Treating $\infty$ as a number — $\lim a_n = +\infty$ says how a sequence diverges, not that it has a limit.
Summing a series without checking $|q| < 1$ — for $q = 2$ the formula returns a number that is the total of nothing.
Assuming the terms must reach the limit — $\frac{1}{n}$ has limit $0$ although no term of it is zero.
Cancelling $\frac{n^2}{n^3}$ down to one — dividing by $n^3$ leaves $\frac{1}{n}$ , which tends to zero.
Evaluating $1^\infty$ as $1$ — that is an indeterminate form; the limit of $\left(1 + \frac{1}{n}\right)^n$ is $e$ , not $1$ .
Writing $0.(9) \approx 1$ — there is no approximation here; both notations name the same number.

Formula card

Topic: Limit of a sequence and the geometric series

Limit of a sequence
$\lim_{n \to \infty} a_n = g$
a sequence convergent to the number g
Definition of the limit
$\forall \varepsilon > 0 \;\; \exists N \;\; \forall n > N : \; |a_n - g| < \varepsilon$
from some point on, every term lies closer to g than epsilon
The basic limit
$\lim_{n \to \infty} \frac{1}{n} = 0$
every quotient limit is derived from it
Quotient of polynomials
$\lim_{n \to \infty} \frac{a n^k + \ldots}{b n^k + \ldots} = \frac{a}{b}$
for equal degrees — the ratio of the leading coefficients
The number e
$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828$
a limit that is the definition of a constant
Sum of a geometric series
$S = \frac{a_1}{1 - q}, \qquad |q| < 1$
only for a convergent one — otherwise there is no sum

The sequence a(n) = 2 + 1/n approaches 2 without ever reaching it. The horizontal helper line is the limit: from some point on, every dot lies arbitrarily close to it.

Partial sums of the series 1 + 1/2 + 1/4 + … Each one adds half of what is still missing from 2 — which is why infinitely many terms add up to a finite total.

Frequently asked questions

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min