Sequences

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A sequence is a list of numbers written in a definite order denoted as:

$$a_1,a_2,...,a_n,...=\{a_n\} ,$$

where $a_1$ is the first term of the sequence, $a_2$ is the second term and in general, $a_n$ is the nth term. (Note that often, unless stated otherwise, it is assumed that the values of $n$ start with $n=1$).

For example,

$$5,10,15,20,25...= {5n} = {a_n}$$ or $$-\frac{1}{3},\frac{1}{9},-\frac{1}{27},\frac{1}{81}...=\left\{\frac{1}{\left(-3\right)^{n}}\right\}_{n=1}^{\infty}$$

If a sequence ${a_n}$ has a limit, that is $\lim_{n\to\infty}a_n=L$ where $L$ is a real number, then it is said to be convergent, and converges to $L$.

If the limit of the sequence does not exist, then the sequence is said to be divergent.

For instance, the sequence $\{r^{n}\}$is convergent if $-1<r\leq1$, that is, $\lim_{n\to\infty}r^n=\begin{cases}0&,\quad\mathrm{if}-1<r<1\\1&,\quad\mathrm{if} r=1\end{cases}$, and divergent otherwise.

Example

Determine if each sequence is convergent or divergent.

1. $\{a_n\}=\{3+e^{-n}\}$
2. $\{a_n\}=\{(-0.5)^n\}$
3. $\{a_n\}=\left\{\frac{n^2}{3+5n}\right\}$
4. $\{a_n\}=\left\{(-1)^n\right\}$

Solution

1. $\lim_{n\to\infty}a_n=\lim_{n\to\infty}(3+e^{-n})=3.$ Convergent.
2. $\operatorname*{lim}_{n\to\infty}a_{n}=\operatorname*{lim}_{n\to\infty}(-0.5)^{n}=0 .$ Convergent.
3. $\lim_{n\to\infty}a_{n}=\lim_{n\to\infty}\frac{n^{2}}{3+5n}=\lim_{n\to\infty}\frac{n}{3/n+5}=+\infty.$ Divergent.
4. Note that $\{a_{n}\}=\left\{(-1)^{n}\right\}=-1,1,-1,1,...$. $\{a_n\}=\left\{(-1)^n\right\}$ does not exist. Divergent.