Conditional Convergence

Conditional Convergence

Describes a series that converges but does not converge absolutely. That is, a convergent series that will become a divergent series if all negative terms are made positive.

Example: 1 - 1/2 + 1/3 - 1/4 + … converges conditionally; but 1 + 1/2 + 1/3 + 1/4 + … diverges (harmonic series).

See also

Convergence tests, harmonic series

Key Formula

\sum_{n=1}^{\infty} a_n \text{ converges, but } \sum_{n=1}^{\infty} |a_n| \text{ diverges}

Where:

$a_n$ = The nth term of the series, which must include both positive and negative values
$|a_n|$ = The absolute value of the nth term
$\sum_{n=1}^{\infty}$ = The sum of all terms from n = 1 to infinity

Worked Example

Problem: Determine whether the alternating harmonic series converges conditionally, converges absolutely, or diverges.

Step 1: Write out the series. The alternating harmonic series is:

\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots

Step 2: Test whether the series converges using the Alternating Series Test. The terms

b_n = \frac{1}{n}

are positive, decreasing, and approach 0 as

n \to \infty

. All three conditions are met, so the series converges.

b_n = \frac{1}{n} > 0, \quad b_{n+1} < b_n, \quad \lim_{n \to \infty} b_n = 0

Step 3: Test for absolute convergence by taking the absolute value of each term. This gives the standard harmonic series:

\sum_{n=1}^{\infty} \left|\frac{(-1)^{n+1}}{n}\right| = \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots

Step 4: The harmonic series is a well-known divergent series (it can be shown to diverge via the integral test or comparison test). Therefore the series does NOT converge absolutely.

\sum_{n=1}^{\infty} \frac{1}{n} = \infty

Step 5: Since the original series converges but the series of absolute values diverges, the alternating harmonic series is conditionally convergent.

Answer: The alternating harmonic series converges conditionally. It converges to ln(2) ≈ 0.693, but the series of absolute values (the harmonic series) diverges.

Another Example

This example contrasts with the first by showing a series that passes the absolute convergence test. The key difference is the $n^2$ in the denominator versus $n$ , which makes the terms shrink fast enough for the absolute-value series to converge too. This helps students distinguish between conditional and absolute convergence.

Problem: Determine whether the series

\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^2}

converges conditionally, converges absolutely, or diverges.

Step 1: Write out the first few terms of the series:

\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2} = -1 + \frac{1}{4} - \frac{1}{9} + \frac{1}{16} - \cdots

Step 2: Check whether the series converges. By the Alternating Series Test,

b_n = \frac{1}{n^2}

is positive, decreasing, and approaches 0. The series converges.

Step 3: Now check for absolute convergence. Take the absolute value of each term to get the p-series with p = 2:

\sum_{n=1}^{\infty} \left|\frac{(-1)^n}{n^2}\right| = \sum_{n=1}^{\infty} \frac{1}{n^2}

Step 4: A p-series converges when

p > 1

. Here

p = 2 > 1

, so the series of absolute values converges (it equals

\pi^2/6

). The series converges absolutely, not conditionally.

\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \approx 1.645

Answer: This series converges absolutely, NOT conditionally. Absolute convergence is a stronger condition than conditional convergence.

Frequently Asked Questions

What is the difference between conditional convergence and absolute convergence?

A series is absolutely convergent if the sum of the absolute values of its terms converges. A series is conditionally convergent if it converges, but the sum of the absolute values diverges. Absolute convergence is the stronger condition: every absolutely convergent series also converges, but a conditionally convergent series depends on the cancellation between positive and negative terms.

Can a series with all positive terms be conditionally convergent?

No. If all terms are positive (or all non-negative), then the series equals its own absolute-value series. So it either converges absolutely or diverges — conditional convergence is impossible. A series must have infinitely many positive and infinitely many negative terms to be conditionally convergent.

What is the Riemann Rearrangement Theorem and why does it matter for conditional convergence?

The Riemann Rearrangement Theorem states that if a series is conditionally convergent, you can rearrange its terms to make the series converge to any real number you choose, or even diverge to infinity. This remarkable result shows that the order of terms in a conditionally convergent series matters enormously, unlike in an absolutely convergent series where any rearrangement gives the same sum.

Conditional Convergence vs. Absolute Convergence

	Conditional Convergence	Absolute Convergence
Definition	Series converges, but the series of absolute values diverges	Series of absolute values converges (which guarantees the original series converges too)
Strength	Weaker form of convergence	Stronger form of convergence
Classic example	Alternating harmonic series: $\sum (-1)^{n+1}/n$	Alternating p-series with p > 1: $\sum (-1)^n/n^2$
Rearrangement of terms	Rearranging terms can change the sum or cause divergence	Any rearrangement converges to the same sum
Sign of terms	Must have both positive and negative terms	Can have all positive, all negative, or mixed-sign terms

Why It Matters

Conditional convergence appears in calculus courses when you study infinite series, particularly power series and their intervals of convergence. At endpoints of a power series' interval, you often need to determine whether convergence is conditional or absolute. Understanding the distinction also matters in physics and engineering, where rearranging the terms of a conditionally convergent series can lead to incorrect results — a subtlety that has real consequences in numerical computation.

Common Mistakes

Mistake: Concluding a series is conditionally convergent just because it has alternating signs.

Correction: Alternating signs are necessary but not sufficient. You must verify two things: (1) the series actually converges (e.g., via the Alternating Series Test), and (2) the series of absolute values diverges. If the absolute-value series also converges, the series is absolutely convergent, not conditionally convergent.

Mistake: Forgetting to check absolute convergence after showing a series converges.

Correction: When a problem asks you to classify convergence, always test the absolute-value series. Simply proving convergence does not tell you whether it is conditional or absolute. Test

\sum |a_n|

explicitly — often with a p-series comparison, integral test, or ratio test.

Related Terms

Series — General concept of summing infinitely many terms
Convergent Series — A series whose partial sums approach a finite limit
Absolute Convergence — Stronger form of convergence, contrasted with conditional
Divergent Series — What the absolute-value series does when convergence is conditional
Convergence Tests — Methods used to determine if a series converges
Harmonic Series — Its alternating version is the classic conditionally convergent example