Conditional Convergence and Rearranging Terms

Why Conditional Convergence Matters

A series can converge even when the sum of the magnitudes of its terms does not. In that case, the positive and negative terms balance in a way that makes the original order important. This is called conditional convergence.

That dependence on order has a striking consequence: for a conditionally convergent series, rearranging the terms can change the sum. We will first classify a familiar example, then explain why the positive and negative terms each have unlimited total magnitude, and finally prove how that fact makes rearrangement possible.

Absolute convergence is therefore a stronger property than convergence of the original series. A conditionally convergent series does have a finite sum in its given order, but its terms do not have a finite total magnitude.

A Standard Example: The Alternating Harmonic Series

Consider the series whose terms alternate in sign while their magnitudes are the terms of the harmonic series:

Set \(b_n=1/n\). These positive terms decrease because \(1/(n+1)<1/n\), and they approach zero as \(n\) increases. The Alternating Series Test says that an alternating series with decreasing positive term magnitudes tending to zero converges. Thus the displayed series converges.

For absolute convergence, take the absolute value of each term. The result is the harmonic series, which diverges:

Consequently, the alternating harmonic series is conditionally convergent: it converges in its stated order, but not absolutely.

Worked Example: A Conditional \(p\)-Series

Why Both Signs Must Contribute an Infinite Total

For a real term \(a_n\), define its positive part and its negative magnitude by

Thus \(a_n=a_n^+-a_n^-\) and \(|a_n|=a_n^++a_n^-\). The positive part records a positive term (and is zero for a nonpositive term); the negative part records the magnitude of a negative term (and is zero for a nonnegative term).

Suppose \(\sum a_n\) converges, but \(\sum |a_n|\) diverges. Then the sums of the positive parts and negative magnitudes must both diverge:

Here is why. If the positive parts had a finite sum, their partial sums would be bounded. The partial sums of the original series, which equal the positive-part partial sums minus the negative-part partial sums, converge and are therefore bounded. Subtracting those bounded original partial sums from the bounded positive-part partial sums shows that the negative-part partial sums are bounded too. Both sequences of partial sums are nondecreasing and nonnegative, so boundedness would make both converge. Then their sum, the absolute-value series, would converge—a contradiction. The same reasoning rules out a finite sum of negative magnitudes.

This proves that any conditionally convergent real series has infinitely much positive mass and infinitely much negative mass. Its original convergence comes from a delicate balance between those two contributions.

Rearranging a Conditional Series

A rearrangement changes the order of the terms without omitting or repeating any term. For finite sums, changing order leaves the sum unchanged. For infinite sums, conditional convergence allows something different: there is a rearrangement that converges to any prescribed real number.

The construction uses the two infinite totals just established. List the positive terms in their original order, and list the negative terms in their original order. There is an unlimited total available in each list, while individual terms become arbitrarily small because convergence of the original series implies \(a_n\to0\).

To see why the resulting partial sums approach \(L\), consider a stage that crosses \(L\) from below by adding positive terms. Just before the last term \(p\) is added, the sum is at most \(L\); just after it is added, the sum is greater than \(L\). The overshoot is therefore at most \(p\). A stage crossing from above by adding a negative term has the same property: its undershoot is at most the magnitude of the last term added.

These last terms approach zero. Indeed, the construction uses infinitely many distinct terms, and the terms of the original convergent series approach zero. Thus the errors at the stage endpoints approach zero. Between two endpoints, the partial sums move monotonically from one side of \(L\) to the other, so their distance from \(L\) cannot exceed the larger of the endpoint errors. Hence every partial sum in the rearranged series approaches \(L\), proving the theorem.

The special starting rule for \(L<0\) matters: beginning with positive terms would start at zero, already above \(L\), and would not create the intended first crossing from below. Beginning with a negative stage fixes that issue. For \(L\geq0\), the initial positive stage is valid, including when \(L=0\), because the first positive term takes the sum strictly above zero.

Worked Example: Rearranging the Alternating Harmonic Series

A Common Mistake: Taking the Absolute Value of a Sum

A common error is to treat the absolute value of a sum as though it were the sum of the absolute values. They are generally different. For example,

In general, the triangle inequality gives \(\left|\sum_{n=1}^{N}a_n\right|\leq\sum_{n=1}^{N}|a_n|\), not equality. Therefore, knowing that the original series converges says nothing by itself about whether the absolute-value series converges. For the alternating harmonic series, cancellation helps the original series converge, while removing the signs leaves the divergent harmonic series.

Why the Distinction Matters

When classifying a series, keep the two questions separate: does the series in its given order converge, and does the series of absolute values converge? The first answer concerns the balance of signed terms. The second asks whether the magnitudes themselves have a finite total.

For AP problems, the Alternating Series Test can establish convergence when its hypotheses hold: the term magnitudes decrease and tend to zero. To test for absolute convergence, remove the alternating signs and classify the resulting nonnegative series using an appropriate convergence test. If the original converges and the absolute-value series diverges, the correct classification is conditional convergence.