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单词 ConditionallyConvergentRealSeries
释义

conditionally convergent real series


Theorem.  If the series

u1+u2+u3+(1)

with real terms ui is conditionally convergent, i.e. converges but |u1|+|u2|+|u3|+ diverges,then the both series

a1+a2+a3+and-b1-b2-b3-(2)

consisting of the positive and negative terms of (1) are divergent — more accurately,

limni=1nan=+andlimni=1n(-bn)=-.

Proof.  If both of the series (2) were convergentMathworldPlanetmath, having the sums A and -B, then we had

0|u1|+|u2|++|un|<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. a1+a2+a3+ were convergent, then the partial sum a1+a2++an were below a finite bound for each n, whereas thenth partial sum of the divergent one of (2) would tend to  - as n; then should also the nth partial sum of (1) tend to  -.

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更新时间:2025/5/4 14:07:23