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Theorem 2

Given a series


one can form a new series by listing the terms in a different order, for example

a1 + a3 + a2 + a5 + a4 + ....

Such a series is called a rearrangement of 09_infinite_series-339.gif. The difference between absolute convergence and conditional convergence is shown emphatically by the following pair of theorems.


A.    Every rearrangement of an absolutely convergent series is also convergent and has the same sum.

B.     Let 09_infinite_series-340.gif be a conditionally convergent series.

(i) The series has a rearrangement which diverges to ∞. (ii) The series has another rearrangement which diverges to -∞.

(iii) For each real number r, the series has a rearrangement which converges to r.

We shall not prove these theorems. Instead we give a pair of rearrangements of the conditionally convergent series


one diverging to ∞ and the other converging to - 1. The alternating series


conditionally converges to a number between ½ and 1.

To get a rearrangement which diverges to ∞, we write down terms in the following order:

  • 1st positive term,
  • 1st negative term,
  • next 2 positive terms,
  • 2nd negative term,
  • next 4 positive terms,
  • 3rd negative term,
  • next 2m positive terms,
  • mth negative term,

We thus obtain the series


Each block of 2m positive terms adds up to at least ,


However, all the negative terms except -1/2 and - have absolute value ≤ 1/6. Hence after the mth negative term the partial sum is more than


Therefore the partial sums, and hence the series, diverge to x.

To get a rearrangement which converges conditionally to -1 we proceed as follows:

Write down negative terms until the partial sum is below -1, then positive terms until the partial sum is above -1, then negative terms until the partial sum is below -1, and so on.

The mth time the partial sum goes above -1, it must be between -1 and - 1 + (1/m). The with time it goes below -1 it must be between -1 and -1 -(1/m). Therefore the series converges to -1.

Last Update: 2006-11-08