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\title {Sequences\\Mathematics}
\author {Ivan de Jes'us Pompa Garc'ia}
\date {October 2006}
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\section {Definition}
A succession of real numbers where every term of the form $x_i$ is followed by other terms is called a sequence.\\

A sequence has the form as follows:

\begin{center}$x_1,x_2,x_3,...,x_n,...$\end{center}

and is denoted by

\begin{center}$\{x_n\}$\end{center}

Every term of the sequence is called element or element of the sequence.

\section{Bounded Sequences}
A sequence is said to be bounded when it's possible to find $M>0$ such that:

\begin{center}$|X_n|<M$\end{center}

or it's equivalent inequalty:

\begin{center}$-M<X_n<M$\end{center}

Consider the next example:

\begin{center}$\frac{1}{2},\frac{2}{3},\frac{3}{4},...,\frac{n}{n+1},...$\end{center}

See that no matter how large $n$ is taken, the sequence always remains less that 1, i.e.

\begin{center}$|x_n|=\frac{n}{n+1}<1$\end{center}

so, we can conclude that the sequence $\{\frac{n}{n+1}\}$ is bounded.

\section{Null Sequences}
Let $\epsilon>0$. $\{x_n\}$ it's a null sequence if it's possible to find a number $p>0$ such that:
\begin{center}$|x_n|<\epsilon$\end{center}

and it remains $\forall n$ where $n>p$.

Maybe this can be cleared with the next example:

Let $\{\frac{1}{n}\}$ a sequence. To probe that expression it's a null sequence, we need to find a number $p>0$ such that $n>p$ for all values of $n$. So, doing some algebra we got that:

\begin{center}$\frac{1}{n}<\epsilon$\end{center}

it follows:

\begin{center}${n}>\frac{1}{\epsilon}$\end{center}



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