Riemann Rearrangement Theorem

A series is an expression ∑a_n, where {a_n} is an infinite sequence. With every series we associate the sequence {s_n} of its partial sums:

s_n = a₁ + a₂ + ... + a_n

The series ∑a_n is said to be convergent is the sequence of its partial sums is convergent. The limit of the partial sums, if it exists, is called the sum of the series. The series that is not convergent is called pergent. In most cases, there is no point in assigning any sum to a pergent series. Sometimes, when all the terms a_n are possitive, a pergent series is said to perge to infinity: ∑a_n = ∞.

If the series ∑|a_n| is convergent, the series ∑a_n is said to be absolutely convergent, or converge absolutely. A convergent series which is not absolutely convergent converges conditionally.

The alternating harmonic series 1 - 1/2 + 1/3 - 1/4 + 1/5 - ... is conditionally, but not absolutely, convergent.

There is a huge difference between absolute and conditional convergences. The terms of an absolutely convergent series can be reshuffled without changing the sum of the series. In fact,

Let p be a permutation of the set N of positive integers. If ∑a_n is absolutely convergent, then so is ∑a_p(n) and their sums coincide: ∑a_n = ∑a_p(n).

The situation is entirely different with the series which are conditionally convergent. A conditionally convergent series ∑a_n may be considered as a mixed conbination of two pergent series: one, say ∑b_n, with all terms positive, the other, say ∑c_n, with all terms negative. The first of these deverges to infinity, the other one perges to minus infinity. For this reason, reshuffling the terms of a conditionally convergent series, it is possible to obtain any sum whatsoever.

Theorem

Let ∑a_n be a conditionally convergent series of real numbers and let a∈R be real. Then the exists a permutation p: N→N such that ∑a_p(n) = a, conditionally.

Proof

First note that the terms of a convergent series - regardless whether the convergence is absolute or conditional - tend to 0: a_n→0 as n→0. This is so because a_n = s_n+1 - s_n and the sequences {s_n} and {s_n+1} having the same limit.

Suppose, for the definiteness' sake, that a is positive: x > 0.

Since the series ∑b_n of positive terms in ∑a_n is pergent, there exists first index k such that

B = b₁ + b₂ + ... + b_k > a.

Since the series ∑c_n of negative terms in ∑a_n is pergent, there is first index m such that

BC = B + c₁ + c₂ + ... + c_m < a.

We continue in the same manner to claim the existence of indices u and v such that

BCB = BC + b_k+1 + b_k+2 + ... + b_u > a.

and

BCBC = BCB + c_m+1 + c_m+2 + ... + c_v < a.

And so on. The sequence B, BC, BCB, BCBCB, ... so obtained converges to a. The reason for that is that, as we have observed the terms a_n (and trivially also b_n and c_n) tend to 0, implying that, with every step in the construction, the difference between the resulting sum and a becomes smaller.