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Divergent Sequence

Divergent Sequence

A sequence that does not converge. For example, the sequence 1, 2, 3, 4, 5, 6, 7, ... diverges since its limit is infinity (∞). The limit of a convergent sequence must be a real number.

 

 

See also

Diverge, divergent series, converge, convergent series

Key Formula

A sequence {an} diverges if there is no real number L such that lim⁡n→∞an=L.\text{A sequence } \{a_n\} \text{ diverges if there is no real number } L \text{ such that } \lim_{n \to \infty} a_n = L.
Where:
  • ana_n = The nth term of the sequence
  • LL = A proposed finite limit (a real number)
  • nn = The index of the term, increasing toward infinity

Worked Example

Problem: Determine whether the sequence given by a_n = (-1)^n is convergent or divergent.
Step 1: Write out the first several terms of the sequence to observe its behavior.
a1=−1,  a2=1,  a3=−1,  a4=1,  a5=−1,…a_1 = -1,\; a_2 = 1,\; a_3 = -1,\; a_4 = 1,\; a_5 = -1, \ldots
Step 2: Notice that the terms alternate between −1 and 1 forever. The sequence never settles toward a single value.
Step 3: For convergence, we would need a real number L such that the terms get arbitrarily close to L. If L = 1, the odd-indexed terms are always at distance 2 from L. If L = −1, the even-indexed terms are always at distance 2 from L. No single real number works.
∣an−L∣↛0 for any real L|a_n - L| \not\to 0 \text{ for any real } L
Step 4: Since no finite limit exists, the sequence diverges.
Answer: The sequence a_n = (-1)^n is divergent because it oscillates between −1 and 1 and never approaches a single real number.

Another Example

Problem: Determine whether the sequence a_n = n^2 converges or diverges.
Step 1: Write out the first few terms.
a1=1,  a2=4,  a3=9,  a4=16,  a5=25,…a_1 = 1,\; a_2 = 4,\; a_3 = 9,\; a_4 = 16,\; a_5 = 25, \ldots
Step 2: The terms grow without bound. For any large number M you choose, eventually every term exceeds M.
lim⁡n→∞n2=∞\lim_{n \to \infty} n^2 = \infty
Step 3: Since infinity is not a real number, the sequence has no finite limit and therefore diverges.
Answer: The sequence a_n = n^2 is divergent because it grows without bound.

Frequently Asked Questions

Does a divergent sequence always go to infinity?
No. A sequence can diverge without going to infinity. For example, the sequence (-1)^n bounces between −1 and 1 forever. It diverges because it never settles on a single value, even though it stays bounded. Divergence simply means the sequence does not converge to any one real number.
How can you tell if a sequence is divergent?
Try to find the limit of a_n as n approaches infinity. If the limit does not exist (the terms oscillate), or if the limit is infinity or negative infinity, the sequence is divergent. Common signs include terms that grow without bound, terms that oscillate without settling, or terms that behave erratically.

Divergent Sequence vs. Convergent Sequence

A convergent sequence has terms that approach a specific real number L as n increases. A divergent sequence does not — either the terms grow without bound, oscillate, or otherwise fail to settle near any single finite value. Every sequence is exactly one or the other: convergent or divergent.

Why It Matters

Recognizing divergence is essential when working with limits in calculus and analysis. Many formulas and theorems (such as L'Hôpital's rule applications or series convergence tests) require you to first determine whether a sequence converges or diverges. In real-world modeling, a divergent sequence can signal instability — for instance, if a population model's terms grow without bound, the model may be unrealistic.

Common Mistakes

Mistake: Assuming a bounded sequence must converge.
Correction: A sequence can stay between fixed bounds yet still diverge. The sequence (-1)^n is bounded between −1 and 1 but diverges because it oscillates. Boundedness alone does not guarantee convergence; the terms must also approach a single value.
Mistake: Confusing a divergent sequence with a divergent series.
Correction: A divergent sequence refers to the behavior of individual terms a_n. A divergent series refers to the sum a_1 + a_2 + a_3 + ··· having no finite value. These are related but distinct concepts. For example, the sequence 1/n converges to 0, but the series 1 + 1/2 + 1/3 + ··· diverges.

Related Terms

  • Convergent Sequence — The opposite — a sequence that has a finite limit
  • Sequence — The broader concept; an ordered list of numbers
  • Limit — The value a convergent sequence approaches
  • Diverge — General term for failing to converge
  • Divergent Series — A series whose partial sums do not converge
  • Infinity — Where some divergent sequences tend toward
  • Convergent Series — A series whose partial sums form a convergent sequence