A divergent sequence is a sequence in College Algebra whose terms do not approach a single limit. The terms may grow without bound, shrink away from a value, or keep changing without settling.
A divergent sequence in College Algebra is a sequence that does not settle toward one fixed number. Instead of getting closer and closer to a limit, its terms either grow without bound, drop without bound, or keep moving around without approaching anything stable.
The version students see most often in this course is a divergent geometric sequence. That happens when you start with a first term and multiply by a common ratio bigger than 1 each time. For example, if the first term is 2 and the ratio is 3, the sequence goes 2, 6, 18, 54, and so on. Each term is larger than the one before it, so the values spread farther apart from any finite number.
You can write a geometric sequence with the formula a_n = a_1 \cdot r^{n-1}. For a divergent geometric sequence, the ratio r is usually greater than 1, which causes exponential growth. The formula itself does not tell you that a sequence is divergent just because it is geometric, though. You still look at what the terms do as n gets larger.
It also helps to separate the sequence from the series. The sequence is the list of terms, while a series is the sum of those terms. A divergent sequence may show up in a context like compound interest or population growth, but if you try to add infinitely many growing terms, the series will not converge either.
One common mistake is thinking “divergent” always means the terms increase forever. That is only one type. A sequence can also diverge by decreasing without bound or by oscillating without approaching a limit. In College Algebra, the main job is to look at the pattern and decide whether the terms settle down or not.
Divergent sequences show up anytime College Algebra connects repeated multiplication to growth patterns. If you are modeling money, population, or any process that scales by the same factor each step, you need to know whether the sequence stays bounded or keeps expanding. That choice changes how you interpret the numbers and whether a long-term value exists.
This term also gives you practice with limits in a very concrete way. A sequence is not just a list of numbers, it is a pattern, and you are often asked to tell what happens as n gets large. If the terms do not approach a specific value, then there is no limit to report. That idea carries into later work with exponential functions and series.
In problem sets, divergent sequences often appear in geometric sequence questions, recursive formulas, and word problems. You may be asked to identify the common ratio, write the nth term, or explain why the pattern cannot converge. Knowing what divergence looks like helps you avoid treating every geometric pattern as if it had a finite endpoint.
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view galleryGeometric Sequence
A divergent sequence in College Algebra is often a geometric sequence with a ratio greater than 1. In that case, each term is formed by multiplying the previous term by the same factor, so the values grow faster and faster. If the ratio is between 0 and 1, the same geometric structure can behave very differently and may converge instead.
Common Ratio
The common ratio tells you what happens from one term to the next in a geometric sequence. When that ratio is greater than 1, the sequence usually diverges upward because each term is bigger than the last by a fixed factor. Looking at the ratio is one of the fastest ways to predict the long-term behavior of the sequence.
Convergent Sequence
Convergent sequence is the opposite idea. Instead of moving away from every fixed value, the terms get closer to one limit. Comparing divergent and convergent sequences helps you decide whether a pattern has a stable long-term value or keeps escaping it.
a_n
The notation a_n labels the nth term of a sequence, and it is the piece you usually examine when deciding whether the sequence diverges. If a_n keeps getting larger without bound or does not settle to a single number, the sequence is divergent. This notation shows up in formulas, tables, and graph interpretations.
A quiz or problem set question may give you a sequence, a recursive rule, or a graph and ask whether it is divergent. Your job is to check the pattern, often by finding the common ratio, writing the explicit formula, or describing what happens as n increases. For a geometric sequence, seeing r > 1 is a quick clue that the terms grow without bound.
You may also be asked to compare divergence with convergence. In those questions, do not just say “it gets bigger,” because a sequence can diverge in more than one way. Use the actual pattern of terms, then state whether a limit exists. If the sequence is tied to a real situation, explain the growth in context, such as repeated multiplication in an interest or population model.
These are easy to mix up because both describe long-term behavior of sequences. A convergent sequence approaches one specific value, while a divergent sequence does not. In College Algebra, the quickest check is to ask whether the terms settle near a limit or keep moving away from any fixed number.
A divergent sequence is a sequence whose terms do not approach one finite limit.
In College Algebra, the most common example is a geometric sequence with common ratio greater than 1.
Use the nth-term formula a_n = a_1 \cdot r^{n-1} to track how the terms change as n grows.
A sequence can diverge by growing without bound, shrinking without bound, or failing to settle at all.
Do not confuse a divergent sequence with a divergent series, since one is a list of terms and the other is a sum.
It is a sequence whose terms do not approach a single limit as the index increases. In many College Algebra examples, the terms keep growing because each term is multiplied by the same number greater than 1. That means the pattern does not settle down.
Check the common ratio. If the ratio is greater than 1, the terms increase by repeated multiplication and usually grow without bound. If the ratio is between 0 and 1, the sequence may converge instead, so the ratio matters a lot.
No. A sequence is the list of terms, while a series is the sum of those terms. A divergent sequence does not approach a limit, and if you try to add infinitely many terms from a growing sequence, that series will also fail to converge.
Yes. Divergence does not always mean the terms get larger. A sequence can also diverge by dropping without bound or by bouncing around without approaching one value. The main idea is that it does not settle at a limit.