14 - Sequences

Real Sequence

A real sequence or sequence of real numbers is a function

$$\alpha :\mathbb{N}\to \mathbb{R}$$

where $\alpha (n)$ is the $n^{\text{th}}$ term of the sequence
Then you can also represent the sequence as

$$(a_n{)}_{n\ge 1}=(a_n{)}_{n=1}^{\infty }=a_1,a_2,\cdots ,$$

Complex Sequence

Similar to Real Sequence but with

$$a:\mathbb{N}\to \mathbb{C}\text{ and }{a}_n\in \mathbb{C}\text{ for }n\ge 1$$

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Note that the order of the terms in a sequence are very important!

Notation on sequences

$$\begin{array}{rl}(a_n)& \text{ represents the sequence}\\ a_n& \text{ represents the term of the sequence}\end{array}$$

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Tail of a Sequence

Let $(a_n)$ be a sequence then
Tail of $(a_n)$ is a sequence $(b_n)$, for some natural number $k$

$$b_n=a_{n+k}\text{ for }n\ge 1$$

TLDR: $(b_n)$ is obtained by deleting the first $k$ terms of $(a_n)$