31 lines
1.4 KiB
TeX
31 lines
1.4 KiB
TeX
\section{Absolute Convergence}
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\begin{definition}
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Let $X:=(x_n)$ be a sequence in $\R$. We say that the series $\sum x_n$ is \textbf{absolutely convergent} if the series $\sum |x_n|$ is convergent in $\R$. A series is said to be \textbf{conditionally} ( or \textbf{nonabsolutely}) \textbf{convergent} if it is convergent, but it is not absolutely convergent.
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\end{definition}
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\begin{theorem}
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If a series in $\R$ is absolutely convergent, then it is convergent.
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\end{theorem}
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\begin{theorem}
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If a series $\sum x_n$ is convergent, then any series obtained from it by grouping the terms is also convergent and to the same value.
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\end{theorem}
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\begin{definition}
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A series $\sum y_k$ in $\R$ is a \textbf{rearrangement} of a series $\sum x_n$ if there is a bijection $f$ of $\N$ onto $\N$ such that $y_k=x_{f(k)}$ for all $k \in \N$.
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\end{definition}
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\begin{theorem}[\textbf{Rearrangement Theorem}]
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Let $\sum x_n$ be an absolutely convergent series in $\R$. Then any rearrangement $\sum y_k$ of $\sum x_n$ converges to the same value.
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\end{theorem}
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\begin{theorem}
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If $\sum a_n$ is conditionally convergent, then there exists a rearrangement of $\sum a_n$ such that
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\begin{enumerate}
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\item The rearrangement converges to any real number $\alpha$
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\item The rearrangement diverges to $\pm \infty$
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\item The rearrangement oscillates between any two real numbers.
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\end{enumerate}
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\end{theorem}
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