35 lines
1.6 KiB
TeX
35 lines
1.6 KiB
TeX
\section{Properly Divergent Sequences}
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\begin{definition}
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Let $(x_n)$ be a sequence of real numbers.
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\begin{enumerate}
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\item We say that $(x_n)$ \textbf{tends to} $\pm \infty$, and write $\lim (x_n) = +\infty$, if for every $\alpha \in \R$ there exists a natural number $K(\alpha)$ such that if $n \geq K(\alpha)$, then $x_n > \alpha$.
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\item We say that $(x_n)$ \textbf{tends to} $-\infty$, and write $\lim (x_n) = -\infty$, if for every $\beta \in \R$ there exists a natural number $K(\beta)$ such that if $n \geq K(\beta)$, then $x_n < \beta$.
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\end{enumerate}
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We say that $(x_n)$ is \textbf{properly divergent} in case we have either $\lim (x_n)=+\infty$, or $\lim (x_n)=-\infty$.
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\end{definition}
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\begin{theorem}
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A monotone sequence of real numbers is properly divergent if and only if it is unbounded.
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\begin{enumerate}
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\item If $(x_n)$ is an unbounded increasing sequence, then $\lim (x_n)=+\infty$.
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\item If $(x_n)$ is an unbounded decreasing sequence, then $\lim (x_n) = -\infty$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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Let $(x_n)$ and $(y_n)$ be two sequences of real numbers and suppose that
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\[x_n \leq y_n\ \ \ \text{for all}\ \ \ n \in \N\]
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\begin{enumerate}
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\item If $\lim (x_n) = +\infty$, then $\lim (y_n)=+\infty$.
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\item If $\lim (y_n) = -\infty$, then $\lim (x_n)=-\infty$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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Let $(x_n)$ and $(y_n)$ be two sequences of positive real numbers and suppose that for some $L \in \R, L>0$, we have
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\[\lim (x_n/y_n)=L\]
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Then $\lim (x_n)=+\infty$ if an only if $\lim (y_n)=+\infty$.
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\end{theorem}
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