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conditionally convergent real series
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(Theorem)
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Theorem. If the series
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(1) |
with real terms $u_i$ is conditionally convergent, i.e. converges but $|u_1|\!+\!|u_2|\!+\!|u_3|\!+\cdots$ diverges, then the both series
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(2) |
consisting of the positive and negative terms of (1) are divergent -- more accurately, $$\lim_{n\to\infty}\sum_{i=1}^na_n \;=\; +\infty\quad\mbox{and}\quad\lim_{n\to\infty}\sum_{i=1}^n(-b_n) \;=\; -\infty.$$
Proof. If both of the series (2) were convergent, having the sums $A$ and $-B$ , then we had $$0 \leqq |u_1|\!+\!|u_2|\!+\ldots+\!|u_n| < A\!+\!B$$ for every $n$ . This would however mean that (1) would converge absolutely, contrary to the conditional convergence. If, on the other hand, one of the series (2) were convergent and the other divergent, then we can see that (1) had to diverge, contrary to what is supposed in the theorem. In fact, if e.g. $a_1\!+\!a_2\!+\!a_3\!+\ldots$ were convergent, then the partial sum $a_1\!+\!a_2\!+\ldots+\!a_n$ were below a finite bound for each $n$ , whereas the $n^\mathrm{th}$ partial sum of the divergent one of (2) would tend to $-\infty$ as $n \to \infty$ ; then should also the $n^\mathrm{th}$ partial sum of (1) tend to $-\infty$ .
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"conditionally convergent real series" is owned by pahio.
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Cross-references: bound, finite, partial sum, conditional convergence, sums, convergent, proof, divergent, negative, positive, diverges, converges, conditionally convergent, real, series, theorem
There is 1 reference to this entry.
This is version 4 of conditionally convergent real series, born on 2009-01-03, modified 2009-01-03.
Object id is 11455, canonical name is ConditionallyConvergentSeries.
Accessed 272 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) | | | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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