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Revision difference : non-existence of universal series convergence criterion |
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There exist many criteria for examining the convergence and \PMlinkescapetext{divergence} of series with positive \PMlinkescapetext{terms}. \,They all are sufficient but not necessary. \,It has also been asked whether there would be any criterion which were both sufficient and necessary. \,The famous mathematician Niels Henrik Abel took this question under consideration and proved the
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There exist many criteria for examining the convergence and \PMlinkescapetext{divergence} of series with positive terms. \,They all are sufficient but not necessary. \,It has also been asked whether there would be any criterion which were both sufficient and necessary. \,The famous mathematician Niels Henrik Abel took this question under consideration and proved the
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| \begin{thmplain} |
\begin{thmplain} |
| There exists no such sequence of positive |
There exists no such sequence of positive |
| \begin{align} |
\begin{align} |
| \varrho_1,\,\varrho_2,\,\varrho_3,\,... |
\varrho_1,\,\varrho_2,\,\varrho_3,\,... |
| \end{align} |
\end{align} |
| numbers that a series \,$a_1+a_2+a_3+...$\, with positive \PMlinkescapetext{terms} were convergent always when the condition |
numbers that a series \,$a_1+a_2+a_3+...$\, with positive \PMlinkescapetext{terms} were convergent always when the condition |
| $$\lim_{n\to\infty}\varrho_n a_n = 0$$ |
$$\lim_{n\to\infty}\varrho_n a_n = 0$$ |
| is true but divergent if it is false. |
is true but divergent if it is false. |
| \end{thmplain} |
\end{thmplain} |
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