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,


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


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 the nth partial sum of the divergent one of (2) would tend to  - as n; then should also the nth partial sum of (1) tend to  -.

Title conditionally convergent real series
Canonical name ConditionallyConvergentRealSeries
Date of creation 2013-03-22 18:41:41
Last modified on 2013-03-22 18:41:41
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Theorem
Classification msc 26A06
Classification msc 40A05
Related topic SumOfSeriesDependsOnOrder