Limits of Sequences

Infinite Limits

Definition Limit equal to $+\infty$

We say that a sequence $(u_n)$ has a limit of $+\infty$ if every interval $(A, +\infty)$ (where $A$ is a real number) contains all terms of the sequence from a certain index onwards. We write:$$\lim_{n \to +\infty} u_n = +\infty$$

Graphical Illustration

No matter how large the real number $A$ is, we can find a natural number $n_0$ such that for all $n \ge n_0$, $u_n > A$.
In intuitive terms: "no matter how high the horizontal barrier $A$ is placed, the terms $u_n$ eventually manage to stay above it permanently."

Example

The sequences $(n)$, $(n^2)$, $(\sqrt{n})$, and $(e^n)$ all have limit $+\infty$.

Definition Limit equal to $-\infty$

We say that a sequence $(u_n)$ has a limit of $-\infty$ if every interval $(-\infty, A)$ (where $A$ is a real number) contains all terms of the sequence from a certain index onwards. We write:$$\lim_{n \to +\infty} u_n = -\infty$$

Graphical Illustration

No matter how small (negative) the real number $A$ is, we can find a natural number $n_0$ such that for all $n \ge n_0$, $u_n < A$.In intuitive terms: "no matter how low the horizontal barrier $A$ is placed, the terms $u_n$ eventually manage to stay below it permanently."

Example

The sequences $(-2n)$, $(-n^2)$ and $(-5\sqrt{n})$ all have limit $-\infty$.

Finite Limits and Convergence

Definition Limit equal to a Real Number $\ell$

We say that a sequence $(u_n)$ converges to a real number $\ell$ if every open interval containing $\ell$ contains all the terms from some index onwards. We write:$$\lim_{n \to +\infty} u_n = \ell$$

We say that $\ell$ is the limit of the sequence $(u_n)$.

Definition Divergent Sequence

A sequence that does not have a finite limit is called a divergent sequence. This includes sequences that tend to $\pm\infty$ and sequences that have no limit at all.

Example

The sequence defined by $u_n = (-1)^n$ has no limit. It alternates between $-1$ and $1$ and never settles toward a specific value. It is therefore a divergent sequence.

Operations on Limits

Proposition Sum and Product Rules

Let $\ell$ and $\ell'$ be real numbers.

Sum:

$\displaystyle \lim_{n \to +\infty} u_n$	$\ell$	$\ell$	$+\infty$	$-\infty$	$+\infty$	$-\infty$
$\displaystyle \lim_{n \to +\infty} v_n$	$\ell'$	$+\infty$	$+\infty$	$-\infty$	$-\infty$	$+\infty$
$\displaystyle \lim_{n \to +\infty} (u_n+v_n)$	$\ell+\ell'$	$+\infty$	$+\infty$	$-\infty$	IF	IF

Product:

$\displaystyle \lim_{n \to +\infty} u_n$	$\ell$	$\ell \neq 0$	$\pm\infty$	$0$
$\displaystyle \lim_{n \to +\infty} v_n$	$\ell'$	$\pm\infty$	$\pm\infty$	$\pm\infty$
$\displaystyle \lim_{n \to +\infty} (u_n \times v_n)$	$\ell\ell'$	$\pm\infty$ (sign rule)	$\pm\infty$ (sign rule)	IF

IF stands for Indeterminate Form (Forme Indéterminée).

Proposition Quotient Rule

Let $(u_n)$ and $(v_n)$ be two sequences such that for all $n$, $v_n \neq 0$.

Case where $\lim v_n = \ell' \neq 0$ or $\pm\infty$

$\displaystyle \lim_{n \to +\infty} u_n$	$\ell$	$\ell$	$+\infty$	$+\infty$	$-\infty$	$-\infty$	$\pm\infty$
$\displaystyle \lim_{n \to +\infty} v_n$	$\ell' \neq 0$	$\pm\infty$	$\ell'>0$	$\ell'<0$	$\ell'>0$	$\ell'<0$	$\pm\infty$
$\displaystyle \lim_{n \to +\infty} \left(\frac{u_n}{v_n}\right)$	$\frac{\ell}{\ell'}$	$0$	$+\infty$	$-\infty$	$-\infty$	$+\infty$	IF

Case where $\lim v_n = 0$

$\displaystyle \lim_{n \to +\infty} u_n$	$\ell>0$ or $+\infty$	$\ell<0$ or $-\infty$	$\ell>0$ or $+\infty$	$\ell<0$ or $-\infty$	$0$
$\displaystyle \lim_{n \to +\infty} v_n$	$0$ (pos.)	$0$ (pos.)	$0$ (neg.)	$0$ (neg.)	$0$
$\displaystyle \lim_{n \to +\infty} \left(\frac{u_n}{v_n}\right)$	$+\infty$	$-\infty$	$-\infty$	$+\infty$	IF

Proposition Limits of Arithmetic Sequences

Let $(u_n)$ be an arithmetic sequence with common difference $d$.

If $\boldsymbol{d > 0}$, then $\displaystyle \lim_{n \to +\infty} u_n = +\infty$.
If $\boldsymbol{d < 0}$, then $\displaystyle \lim_{n \to +\infty} u_n = -\infty$.

Proof

An arithmetic sequence is defined by its general term: $u_n = u_0 + n \times d$.
To find the limit, we apply the rules for operations on limits:

Case $d > 0$:
- We know the reference limit: $\displaystyle \lim_{n \to +\infty} n = +\infty$.
- Since $d$ is a strictly positive constant, by the product rule: $$\displaystyle \lim_{n \to +\infty} (n \times d) = +\infty$$
- Adding the constant $u_0$, by the sum rule: $$\displaystyle \lim_{n \to +\infty} (u_0 + nd) = +\infty$$
Case $d < 0$:
- We know: $\displaystyle \lim_{n \to +\infty} n = +\infty$.
- Since $d$ is a strictly negative constant, by the product rule: $\displaystyle \lim_{n \to +\infty} (n \times d) = -\infty$.
- Adding the constant $u_0$, by the sum rule: $\displaystyle \lim_{n \to +\infty} (u_0 + nd) = -\infty$.

Example

Let $u_n = 5 + 3n$. Since $d=3 > 0$, $\displaystyle\lim_{n \to +\infty} u_n = +\infty$.
Let $v_n = 10 - 2n$. Since $d=-2 < 0$, $\displaystyle\lim_{n \to +\infty} v_n = -\infty$.

Comparison and Squeeze Theorems

Theorem Comparison for Infinite Limits

Let $(u_n)$ and $(v_n)$ be two sequences.
If (1) from a certain index $n_0$ onwards, $u_n \ge v_n$, and if (2) $\displaystyle \lim_{n \to +\infty} v_n = +\infty$,
then:$$\lim_{n \to +\infty} u_n = +\infty$$

Theorem Squeeze Theorem / Sandwich Theorem

Let $(u_n)$, $(v_n)$, and $(w_n)$ be three sequences.
If (1) from a certain index onwards, $\textcolor{colordef}{v_n} \leqslant \textcolor{olive}{u_n} \leqslant \textcolor{colorprop}{w_n}$, and (2) $\displaystyle \lim_{n \to +\infty} v_n = \lim_{n \to +\infty} w_n = \ell$,
then the sequence $(u_n)$ converges and $\displaystyle\lim_{n \to +\infty} u_n = \ell$.

Example

Calculate the limit of the sequence $(u_n)$ defined for $n \ge 1$ by $u_n = \dfrac{(-1)^n}{n}$.

Answer

For all $n \ge 1$, we know that $-1 \le (-1)^n \le 1$.
Since $n > 0$, we can divide all parts of the inequality by $n$ without changing the signs:$$-\frac{1}{n} \le \frac{(-1)^n}{n} \le \frac{1}{n}$$We know that:

$\displaystyle \lim_{n \to +\infty} -\frac{1}{n} = 0$
$\displaystyle \lim_{n \to +\infty} \frac{1}{n} = 0$

According to the Squeeze Theorem, since $u_n$ is trapped between two sequences that converge to 0, we conclude:$$\lim_{n \to +\infty} \frac{(-1)^n}{n} = 0$$

Limits of Geometric Sequences

Proposition Limits of Geometric Sequences

Let $q$ be a real number. The limit of the sequence $(q^n)$ depends on the value of $q$:

If $\boldsymbol{q > 1}$, then $\displaystyle \lim_{n \to +\infty} q^n = +\infty$.
If $\boldsymbol{-1 < q < 1}$, then $\displaystyle \lim_{n \to +\infty} q^n = 0$.
If $\boldsymbol{q = 1}$, then the sequence is constant and $\displaystyle \lim_{n \to +\infty} q^n = 1$.
If $\boldsymbol{q \le -1}$, then the sequence has no limit (it diverges).

Proof

($q > 1$):
We use Bernoulli's Inequality: for all $a > 0$ and $n \in \mathbb{N}$, $(1+a)^n \ge 1 + na$.
Let $q = 1 + a$ where $a = q - 1 > 0$.
Then $q^n = (1 + a)^n \ge 1 + na$.
We know that $\displaystyle \lim_{n \to +\infty} (1 + na) = +\infty$ because $a > 0$.
By the Comparison Theorem, since $q^n \ge 1 + na$, we conclude:$$\lim_{n \to +\infty} q^n = +\infty$$
($-1 < q < 1$):
If $q = 0$, the limit is obviously 0.
If $0 < |q| < 1$, let $Q = \frac{1}{|q|}$. Then $Q > 1$.
From the previous proof, we know $\displaystyle \lim_{n \to +\infty} Q^n = +\infty$.
Since $|q|^n = \frac{1}{Q^n}$, then $\displaystyle \lim_{n \to +\infty} |q|^n = 0$.
By the Squeeze Theorem (-$|q|^n\leq q^n\leq |q|^n$), we conclude:$$\lim_{n \to +\infty} q^n = 0$$

Example

$\displaystyle \lim_{n \to +\infty} 1.05^n = +\infty$ because $1.05 > 1$.
$\displaystyle \lim_{n \to +\infty} \left(\frac{1}{2}\right)^n = 0$ because $-1 < 0.5 < 1$.
The sequence $((-2)^n)$ has no limit because $-2 \le -1$.

\(\displaystyle \lim_{n \to +\infty} u_n\)	\(\ell\)	\(\ell\)	\(+\infty\)	\(-\infty\)	\(+\infty\)	\(-\infty\)
\(\displaystyle \lim_{n \to +\infty} v_n\)	\(\ell'\)	\(+\infty\)	\(+\infty\)	\(-\infty\)	\(-\infty\)	\(+\infty\)
\(\displaystyle \lim_{n \to +\infty} (u_n+v_n)\)	\(\ell+\ell'\)	\(+\infty\)	\(+\infty\)	\(-\infty\)	IF	IF

\(\displaystyle \lim_{n \to +\infty} u_n\)	\(\ell\)	\(\ell \neq 0\)	\(\pm\infty\)	\(0\)
\(\displaystyle \lim_{n \to +\infty} v_n\)	\(\ell'\)	\(\pm\infty\)	\(\pm\infty\)	\(\pm\infty\)
\(\displaystyle \lim_{n \to +\infty} (u_n \times v_n)\)	\(\ell\ell'\)	\(\pm\infty\) (sign rule)	\(\pm\infty\) (sign rule)	IF

\(\displaystyle \lim_{n \to +\infty} u_n\)	\(\ell\)	\(\ell\)	\(+\infty\)	\(+\infty\)	\(-\infty\)	\(-\infty\)	\(\pm\infty\)
\(\displaystyle \lim_{n \to +\infty} v_n\)	\(\ell' \neq 0\)	\(\pm\infty\)	\(\ell'>0\)	\(\ell'<0\)	\(\ell'>0\)	\(\ell'<0\)	\(\pm\infty\)
\(\displaystyle \lim_{n \to +\infty} \left(\frac{u_n}{v_n}\right)\)	\(\frac{\ell}{\ell'}\)	\(0\)	\(+\infty\)	\(-\infty\)	\(-\infty\)	\(+\infty\)	IF

\(\displaystyle \lim_{n \to +\infty} u_n\)	\(\ell>0\) or \(+\infty\)	\(\ell<0\) or \(-\infty\)	\(\ell>0\) or \(+\infty\)	\(\ell<0\) or \(-\infty\)	\(0\)
\(\displaystyle \lim_{n \to +\infty} v_n\)	\(0\) (pos.)	\(0\) (pos.)	\(0\) (neg.)	\(0\) (neg.)	\(0\)
\(\displaystyle \lim_{n \to +\infty} \left(\frac{u_n}{v_n}\right)\)	\(+\infty\)	\(-\infty\)	\(-\infty\)	\(+\infty\)	IF