Definition of Sequence

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$
A sequence is an infinite list of numbers $a_1$, $a_2$, $a_3$, $\dots$. We shall write $(a_n)$ or simply $a_n$, where $n\in \N$.

The following are all examples of sequences.

  1. $1$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\dots$
  2. $0$, $0$, $0$, $0$, $0$, $\dots $
  3. $1$, $0$, $\frac{1}{2}$, $0$, $\frac{1}{3}$, $0$, $\frac{1}{4}$, $0$,$\dots$
  4. $1$, $2$, $3$, $4$, $\dots$
  5. $1$, $-1$, $1$, $-1$, $1$, $\dots$
  6. $n^2$, $n\in \N$,
  7. $2n$, $n\in \N$, is the sequence of even numbers,
  8. $2n-1$, $n\in \N$, is the sequence of odd numbers,
  9. $\cos(n) $, $n\in \N $,
  10. $\displaystyle \frac{3n+\sin n}{(2n+7)(n^2+1)}$, $n\in \N$,
  11. $(-1)^{n+1}$ and $\cos \left((n-1)\pi \right)$, $n\in \N$, give the same sequence. (In fact, the same sequence as in example (5).

We can define a sequence as a function from $\N $ to $\R$. That is, $f:\N \to \R$ is defined by $f(n)=a_n$. This is just an alternative way of viewing the concept of sequences, it is equally as valid as our notion of viewing it as a list.

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