Convergence and Divergence.

Definition 3.6.1   The sequence $(u_n)$ is convergent if there exists a real number $l$ verifying the following property (called Cauchy's inequality):

$$\forall \varepsilon >0, \exists N_0 \in \mathbb{N} \; \vert \; n \geq N_0 \Longrightarrow \begin{vmatrix}u_n -l \end{vmatrix} < \varepsilon$$

Otherwise, the sequence $(u_n)$ is divergent.

Notation: $\underset{n \rightarrow \infty}{\text{lim}} u_n = l$ .

Example 3.6.2   Let $u_n=\frac {n-1}{2n+1}$ ; we prove that $(u_n)$ is convergent and that $\underset{n \rightarrow \infty}{\text{lim}} u_n = \frac 12$

Take any $\varepsilon >0$ ; we look for natural numbers such that $\begin{vmatrix}\frac {n-1}{2n+1} -\frac 12 \end{vmatrix} < \varepsilon$ . We have:

$$\begin{vmatrix}\frac {n-1}{2n+1} -\frac 12 \end{vmatrix} < \varepsilon$$

$$\begin{vmatrix}\frac {2(n-1)}{2(2n+1)} -\frac {2n+1}{2(2n+1)} \end{vmatrix} < \varepsilon$$

$$\begin{vmatrix}\frac {-3}{2(2n+1)} \end{vmatrix} < \varepsilon$$

$$\frac {3}{4n+2} < \varepsilon$$

$$4n+2 > \frac {3}{\varepsilon}$$

$$n > \frac 14 \left( \frac {3}{\varepsilon} -2 \right).$$

Let $N_0$ be the integer part of $\frac 14 \left( \frac {3}{\varepsilon} -2 \right)$ ; then for any natural $n$ such that $n>N_0$ , we have $\vert u_n -1/2 \vert< \varepsilon$ .

Proposition 3.6.3   If a sequence $(a_n)$ is convergent, then it is bounded.

Proof. Let $a$ be the limit of the sequence $(a_n)$ ; by definition 6.1 we have:

$$\forall \varepsilon >0, \exists N_0 \in \mathbb{N} \; \vert \; n \geq N_0 \Longrightarrow \begin{vmatrix}a_n -a \end{vmatrix} < \varepsilon .$$

Thus the set $\{ a_n \; \vert \; n >N_0 \}$ is bounded (below by $a -\varepsilon$ and above by $a+ \varepsilon$ ). As the set $\{ a_n \; \vert \; n \leq N_0 \}$ is finite, it is bounded. The set of all the terms of the sequence $(a_n)$ is the union of two bounded sets, hence it is bounded. $\qedsymbol$

For example, any periodic sequence is bounded, but is divergent.

Theorem 3.6.4   Let $(u_n)$ be an arithmetic sequence whose difference is equal to $d$ .

1. If $d=0$ , the sequence is constant.
2. If $d>0$ , then the sequence increases and has $+\infty$ as its limit.
3. If $d<0$ , then the sequence increases and has $-\infty$ as its limit.

Theorem 3.6.5   Let $(v_n)$ be an geometric sequence whose quotient is equal to $q$ .

1. If $q=1$ , the sequence is constant.
2. If $q=0$ , the sequence is constant from its second term.
3. If $\vert q\vert<1$ , the sequence is convergent and its limit is 0.
4. If $q>1$ , the sequence is divergent; its limit is infinite ($+\infty$ or $-\infty$ according to the sign of the first term).
5. If $q<-1$ , the sequence is divergent and has no limit.
6. If $q=-1$ , the sequence is periodic, with period equal to 2.

Definition 3.6.6   The sequence $(u_n)$ has $+\infty$ as its limit if:

$$\forall A>0, \exists N_0 \in \mathbb{N} \; \vert \; n \geq N_0 \Longrightarrow u_n >A$$

We denote: $\underset{n \rightarrow \infty}{\text{lim}}u_n = +\infty$ .

This formula is called Cauchy's inequality too. Write the corresponding definition for $\underset{n \rightarrow \infty}{\text{lim}}u_n=-\infty$ .