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Diverge

Diverge

To fail to approach a finite limit. There are divergent limits, divergent series, divergent sequences, and divergent improper integrals.

 

 

See also

Convergence tests, converge

Worked Example

Problem: Determine whether the series 1 + 2 + 3 + 4 + ... (the sum of all positive integers) converges or diverges.
Step 1: Write the partial sums. The nth partial sum is the sum of the first n positive integers.
Sn=n(n+1)2S_n = \frac{n(n+1)}{2}
Step 2: Check whether the partial sums approach a finite limit as n grows.
lim⁡n→∞Sn=lim⁡n→∞n(n+1)2=∞\lim_{n \to \infty} S_n = \lim_{n \to \infty} \frac{n(n+1)}{2} = \infty
Step 3: Since the partial sums grow without bound rather than settling on a finite number, the series diverges.
Answer: The series 1 + 2 + 3 + 4 + ... diverges because its partial sums increase to infinity.

Another Example

Problem: Does the sequence a_n = (-1)^n converge or diverge?
Step 1: List the first several terms of the sequence.
a1=−1,  a2=1,  a3=−1,  a4=1,  …a_1 = -1,\; a_2 = 1,\; a_3 = -1,\; a_4 = 1,\; \ldots
Step 2: The terms alternate between -1 and 1 forever. They never settle toward a single value.
Step 3: Because the sequence does not approach any one finite limit, it diverges. Note that divergence does not require going to infinity — oscillating without settling is enough.
Answer: The sequence (-1)^n diverges because it oscillates between -1 and 1 and never approaches a single limit.

Frequently Asked Questions

Does diverge always mean going to infinity?
No. A sequence or series can diverge by oscillating without settling on a value, not just by growing without bound. For example, the sequence (-1)^n diverges even though every term is either -1 or 1. Any failure to approach a single finite limit counts as divergence.
How do you test whether a series diverges?
The quickest first check is the Divergence Test (also called the nth-term test): if the terms of the series do not approach zero, the series must diverge. However, terms approaching zero is necessary but not sufficient for convergence — the harmonic series 1 + 1/2 + 1/3 + ... has terms going to zero yet still diverges. Further tests like the ratio test, comparison test, or integral test can determine divergence in less obvious cases.

Diverge vs. Converge

Converge and diverge are opposites. A sequence, series, or integral converges if it approaches a specific finite value; it diverges if it does not. For example, the series 1 + 1/2 + 1/4 + 1/8 + ... converges to 2, while the series 1 + 1 + 1 + 1 + ... diverges to infinity. Every sequence, series, or improper integral either converges or diverges — there is no middle ground.

Why It Matters

Knowing whether a mathematical expression converges or diverges is fundamental in calculus, physics, and engineering. If a series representing a physical quantity diverges, the model may be invalid or require a different approach. Divergence tests are also central to understanding infinite sums in topics like power series, Taylor series, and Fourier series.

Common Mistakes

Mistake: Thinking that if the terms of a series approach zero, the series must converge.
Correction: Terms going to zero is necessary for convergence but not sufficient. The harmonic series (1 + 1/2 + 1/3 + 1/4 + ...) has terms approaching zero, yet it diverges. You need additional convergence tests to be sure.
Mistake: Assuming divergence only means 'goes to infinity.'
Correction: Divergence includes any failure to reach a finite limit. A sequence that oscillates (like (-1)^n) or a limit that does not exist both count as divergent, even when no term is particularly large.

Related Terms