A real number sequence is an ordered list of real numbers, usually denoted by
\[ a_1,\ a_2,\ a_3,\ \ldots,\ a_n,\ \ldots \]
where \(a_n\) is the general term of the sequence, that is, the term occupying position \(n\). To study a sequence means to understand how its terms behave as the index \(n\) grows larger.
The central notion here is that of the limit of a sequence. As \(n\to+\infty\), the terms \(a_n\) may approach a real number, grow without bound, decrease without bound, or fail to exhibit any limiting behaviour at all. For this reason we distinguish between convergent, divergent and oscillating sequences.
As \(n\to+\infty\), the terms \(a_n\) may approach a real number, grow without bound, decrease without bound, or fail to exhibit any limiting behaviour at all.
For this reason we distinguish between convergent, divergent and oscillating sequences.
We now introduce the fundamental definitions concerning limits of sequences, giving a rigorous account of what convergence, divergence and oscillation mean.
Contents
- What a numerical sequence is
- The limit of a sequence
- Convergent sequences
- Divergent sequences
- Oscillating sequences
- Examples of convergent, divergent and oscillating sequences
What a numerical sequence is
A numerical sequence is a function defined on the set of positive natural numbers and taking values in a number set. In the case of real sequences, it is a function
\[ a:\mathbb{N}_{\ge 1}\to\mathbb{R}. \]
To each positive natural number \(n\) there corresponds one and only one real number \(a(n)\). Rather than writing \(a(n)\), for sequences one almost always uses the notation
\[ a_n. \]
The number \(a_n\) is called the n-th term or general term of the sequence. The sequence itself is then denoted by one of the following:
\[ (a_n)_{n\in\mathbb{N}},\qquad (a_n),\qquad a_n. \]
For instance, the formula
\[ a_n=\frac{1}{n} \]
defines the sequence
\[ 1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\ \ldots \]
Here the first term is \(a_1=1\), the second term is \(a_2=\displaystyle \frac{1}{2}\), the third term is \(a_3=\displaystyle \frac{1}{3}\), and so on.
The crucial point is that a sequence is not merely a set of numbers, but a collection of values arranged in a definite order. For example, the sequences
\[ 1,\ 2,\ 1,\ 2,\ 1,\ 2,\ \ldots \]
and
\[ 2,\ 1,\ 2,\ 1,\ 2,\ 1,\ \ldots \]
take the same values, but in a different order. For this reason they must be regarded as different sequences.
The limit of a sequence
The limit of a sequence describes the behaviour of the terms \(a_n\) as the index \(n\) becomes arbitrarily large, that is, as
\[ n\to+\infty. \]
This point matters: in a sequence the index \(n\) ranges over the natural numbers, so one does not study the behaviour as \(n\to-\infty\), but only as \(n\to+\infty\).
A sequence can behave in several ways. It may approach a real number, it may increase without bound, it may decrease without bound, or it may have no limiting behaviour at all. For this reason three fundamental cases are distinguished:
- convergent sequences, when the terms approach a real number;
- divergent sequences, when the terms tend to \(+\infty\) or to \(-\infty\);
- oscillating sequences, when there is neither a finite limit nor an infinite one.
Writing
\[ \lim_{n\to+\infty}a_n=L \]
means that, as \(n\) increases, the terms \(a_n\) draw indefinitely close to the real number \(L\). The number \(L\), when it exists, is called the limit of the sequence.
For example, taking the sequence
\[ a_n=\frac{1}{n}, \]
the terms are
\[ 1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\ \ldots \]
and they become closer and closer to \(0\). In this case one writes
\[ \lim_{n\to+\infty}\frac{1}{n}=0. \]
The limit, however, should not be understood as a value that the sequence necessarily attains. In the previous example no term of the sequence equals \(0\), since
\[ \frac{1}{n}\neq 0 \]
for every \(n\in\mathbb{N}\). Nevertheless the terms come as close to \(0\) as we please, provided \(n\) is taken sufficiently large.
Convergent sequences
A real sequence \((a_n)\) is said to be convergent if there exists a real number \(L\) such that
\[ \lim_{n\to+\infty}a_n=L. \]
In this case one says that the sequence converges to \(L\), or that \(L\) is the finite limit of the sequence.
The rigorous definition is the following:
\[ \lim_{n\to+\infty}a_n=L \iff \forall \varepsilon>0\ \exists n_\varepsilon\in\mathbb{N}\ :\ \forall n\geq n_\varepsilon \,\, , \,\ |a_n-L|<\varepsilon. \]
This definition must be read carefully. The number \(\varepsilon>0\) represents an arbitrarily small distance from the limit \(L\). To say that
\[ |a_n-L|<\varepsilon \]
indeed means that the term \(a_n\) lies at a distance less than \(\varepsilon\) from \(L\).
The definition therefore states that, once any positive distance \(\varepsilon\) has been chosen, however small, there exists an index \(n_\varepsilon\) such that all the terms of the sequence with index \(n\geq n_\varepsilon\) lie at a distance less than \(\varepsilon\) from \(L\).
In geometric terms, once an open interval centred at \(L\) is fixed,
\[ (L-\varepsilon,L+\varepsilon), \]
there exists an index \(n_\varepsilon\) such that all the subsequent terms of the sequence belong to that interval.
In symbols:
\[ n\geq n_\varepsilon \quad\Longrightarrow\quad a_n\in(L-\varepsilon,L+\varepsilon). \]
It is important to note that the definition does not require all the terms of the sequence to be close to \(L\). The first few terms may even be very far from the limit. What matters is that, from a certain index onwards, all the terms remain arbitrarily close to \(L\).
For example, the sequence
\[ a_n=\frac{n}{n+1} \]
converges to \(1\), because its terms
\[ \frac{1}{2},\ \frac{2}{3},\ \frac{3}{4},\ \frac{4}{5},\ \ldots \]
draw closer and closer to \(1\).
Indeed:
\[ \frac{n}{n+1}=1-\frac{1}{n+1}. \]
The quantity subtracted from \(1\), namely \(\displaystyle \frac{1}{n+1}\), becomes smaller and smaller as \(n\) increases. Hence
\[ \lim_{n\to+\infty}\frac{n}{n+1}=1. \]
Convergent sequences always have a finite real limit. For this reason a convergent sequence is also said to be a sequence that admits a finite limit.
Divergent sequences
A real sequence \((a_n)\) is said to be divergent if its terms do not approach a finite real number, but instead become arbitrarily large or arbitrarily small.
More precisely, a sequence can diverge in two ways:
- it may diverge to \(+\infty\), if its terms become greater than any prescribed positive threshold;
- it may diverge to \(-\infty\), if its terms become smaller than any prescribed negative threshold.
In both cases the symbol \(+\infty\) or \(-\infty\) does not represent a real number. To say that a sequence tends to \(+\infty\) or to \(-\infty\) is to describe a behaviour of its terms, not to indicate a value attained by the sequence.
Sequences diverging to \(+\infty\)
A real sequence \((a_n)\) is said to be divergent to \(+\infty\) if, for any positive number \(M\) one fixes, there exists an index \(n_M\) such that all the subsequent terms of the sequence are greater than \(M\).
Formally:
\[ \lim_{n\to+\infty}a_n=+\infty \iff \forall M>0\ \exists n_M\in\mathbb{N}\ :\ \forall n\geq n_M \,\, , \,\ a_n>M. \]
The definition says that, whatever the positive threshold \(M\) may be, however large, from a certain index onwards all the terms of the sequence exceed that threshold.
For example, the sequence
\[ a_n=n^2 \]
diverges to \(+\infty\), because its terms
\[ 1,\ 4,\ 9,\ 16,\ \ldots \]
become arbitrarily large.
Indeed, having fixed \(M>0\), we want
\[ n^2>M. \]
Since \(n\) is positive, this inequality holds when
\[ n>\sqrt{M}. \]
Choosing \(n_M\in\mathbb{N}\) such that
\[ n_M>\sqrt{M}, \]
for every \(n\geq n_M\) we have
\[ n^2>M. \]
Hence
\[ \lim_{n\to+\infty}n^2=+\infty. \]
Sequences diverging to \(-\infty\)
A real sequence \((a_n)\) is said to be divergent to \(-\infty\) if, for any positive number \(M\) one fixes, there exists an index \(n_M\) such that all the subsequent terms of the sequence are smaller than \(-M\).
Formally:
\[ \lim_{n\to+\infty}a_n=-\infty \iff \forall M>0\ \exists n_M\in\mathbb{N}\ :\ \forall n\geq n_M \,\, , \,\ a_n<-M. \]
The definition says that the terms of the sequence fall below any negative threshold. Here too \(-\infty\) is not a value attained by the sequence; it describes the fact that the terms become arbitrarily small.
For example, the sequence
\[ a_n=-n \]
diverges to \(-\infty\), because its terms
\[ -1,\ -2,\ -3,\ -4,\ \ldots \]
become smaller and smaller.
Indeed, having fixed \(M>0\), we want
\[ -n<-M. \]
Multiplying both sides by \(-1\) reverses the inequality:
\[ n>M. \]
Choosing \(n_M\in\mathbb{N}\) such that
\[ n_M>M, \]
for every \(n\geq n_M\) we have
\[ n>M \]
and therefore
\[ -n<-M. \]
Consequently
\[ \lim_{n\to+\infty}(-n)=-\infty. \]
Divergent sequences are not convergent, since they admit no finite real limit. They do, however, have a definite limiting behaviour: they tend to \(+\infty\) or to \(-\infty\).
Oscillating sequences
A real sequence \((a_n)\) is said to be oscillating if it has no limit, whether finite or infinite.
In other words, a sequence oscillates if it is neither convergent nor divergent to \(+\infty\) or to \(-\infty\). Thus an oscillating sequence does not approach a real number, does not increase without bound and does not decrease without bound.
The simplest case is that of a sequence which oscillates indefinitely between distinct values. For example, the sequence
\[ a_n=(-1)^n \]
takes alternately the values
\[ -1,\ 1,\ -1,\ 1,\ -1,\ 1,\ \ldots \]
and so does not approach a single limiting value.
Indeed, considering only the even indices, we obtain
\[ a_{2k}=(-1)^{2k}=1. \]
Hence
\[ \lim_{k\to+\infty}a_{2k}=1. \]
Considering instead the odd indices, we obtain
\[ a_{2k-1}=(-1)^{2k-1}=-1. \]
Hence
\[ \lim_{k\to+\infty}a_{2k-1}=-1. \]
The same sequence therefore has two subsequences converging to different limits. For this reason the sequence \(((-1)^n)\) cannot be convergent.
Moreover it is bounded, since for every \(n\in\mathbb{N}\) we have
\[ -1\leq (-1)^n\leq 1. \]
Being bounded, it can diverge neither to \(+\infty\) nor to \(-\infty\). It is therefore an oscillating sequence.
This example shows that a bounded sequence need not be convergent. Boundedness rules out divergence to \(+\infty\) or to \(-\infty\), but it does not guarantee the existence of a finite limit.
More generally, a sequence may oscillate because it swings between different values, because it behaves differently along different subsequences, or because it does not settle steadily towards any value, finite or infinite.
Examples of convergent, divergent and oscillating sequences
We now look at a few fundamental examples, useful for recognising the main types of limiting behaviour of a sequence.
Example 1. (Sequence converging to \(0\)). Consider the sequence
\[ a_n=\frac{1}{n}. \]
Its terms are
\[ 1,\ \frac{1}{2},\ \frac{1}{3},\ \frac{1}{4},\ \ldots \]
As \(n\) increases, the denominator grows larger and larger while the numerator stays equal to \(1\). As a result the terms of the sequence become smaller and smaller and approach \(0\).
Hence
\[ \lim_{n\to+\infty}\frac{1}{n}=0. \]
The sequence is therefore convergent.
Example 2. (Sequence converging to \(1\)). Consider the sequence
\[ a_n=\frac{n}{n+1}. \]
We can rewrite the general term as follows:
\[ \frac{n}{n+1} = \frac{n+1-1}{n+1} = 1-\frac{1}{n+1}. \]
Since
\[ \frac{1}{n+1}\to0 \]
as \(n\to+\infty\), we obtain
\[ 1-\frac{1}{n+1}\to1. \]
Therefore
\[ \lim_{n\to+\infty}\frac{n}{n+1}=1. \]
This sequence too is convergent.
Example 3. (Sequence diverging to \(+\infty\)).
Consider the sequence
\[ a_n=2n. \]
Its terms are
\[ 2,\ 4,\ 6,\ 8,\ \ldots \]
and they grow without bound. Indeed, having fixed any number \(M>0\), we look for an index \(n_M\) such that, for every \(n\geq n_M\), we have
\[ 2n>M. \]
This inequality is equivalent to
\[ n>\frac{M}{2}. \]
Choosing \(n_M\in\mathbb{N}\) such that
\[ n_M>\frac{M}{2}, \]
for every \(n\geq n_M\) we have \(2n>M\). Hence
\[ \lim_{n\to+\infty}2n=+\infty. \]
The sequence is therefore divergent to \(+\infty\).
Example 4. (Sequence diverging to \(-\infty\)). Consider the sequence
\[ a_n=-3n. \]
Its terms are
\[ -3,\ -6,\ -9,\ -12,\ \ldots \]
and they become smaller and smaller. Having fixed \(M>0\), we want
\[ -3n<-M. \]
Multiplying both sides by \(-1\) reverses the inequality:
\[ 3n>M. \]
Hence
\[ n>\frac{M}{3}. \]
Choosing \(n_M\in\mathbb{N}\) such that
\[ n_M>\frac{M}{3}, \]
for every \(n\geq n_M\) we have
\[ -3n<-M. \]
Therefore
\[ \lim_{n\to+\infty}(-3n)=-\infty. \]
The sequence is thus divergent to \(-\infty\).
Example 5. (Oscillating sequence). Consider the sequence
\[ a_n=(-1)^n. \]
Its terms are
\[ -1,\ 1,\ -1,\ 1,\ -1,\ 1,\ \ldots \]
The sequence does not approach a single value. Indeed, along the even indices we have
\[ a_{2k}=1, \]
while along the odd indices we have
\[ a_{2k-1}=-1. \]
So two subsequences of the same sequence have different limits:
\[ \lim_{k\to+\infty}a_{2k}=1, \qquad \lim_{k\to+\infty}a_{2k-1}=-1. \]
Consequently the sequence is not convergent.
Moreover it is bounded, since for every \(n\in\mathbb{N}\) we have
\[ -1\leq a_n\leq 1. \]
Hence it diverges neither to \(+\infty\) nor to \(-\infty\). The sequence is therefore oscillating.
Example 6. (A bounded but non-convergent sequence)
Consider the sequence
\[ a_n=\sin\left(\frac{n\pi}{2}\right). \]
Its first terms are
\[ 1,\ 0,\ -1,\ 0,\ 1,\ 0,\ -1,\ 0,\ \ldots \]
Here too the sequence is bounded, since its terms lie in the interval \([-1,1]\). Nevertheless it does not converge, because it periodically takes different values and does not settle around a single limit.
For example, for indices of the form \(4k+1\) we have
\[ a_{4k+1}=1, \]
while for indices of the form \(4k+2\) we have
\[ a_{4k+2}=0. \]
Hence
\[ \lim_{k\to+\infty}a_{4k+1}=1, \qquad \lim_{k\to+\infty}a_{4k+2}=0. \]
The sequence has two subsequences with different limits, so it is not convergent. Being bounded, it cannot diverge to \(+\infty\) or to \(-\infty\). It is therefore oscillating.
To sum up, a sequence can display three main types of behaviour: it may converge to a real number, it may diverge to \(+\infty\) or to \(-\infty\), or it may oscillate. This classification underlies the whole study of limits of sequences.