Unbounded sequence

An unbounded sequence is a Calculus II sequence whose terms do not stay within any fixed size, so they grow past every real bound. That means it cannot converge to a finite limit.

What is unbounded sequence?

An unbounded sequence in Calculus II is a sequence whose terms eventually get larger than every finite number in absolute value. If you keep going far enough out in the sequence, the terms do not settle near one value, they keep moving outside any box you try to put around them.

That is why unboundedness and convergence are opposites here. A convergent sequence has terms that get closer and closer to a finite limit as n grows. An unbounded sequence does not have that behavior, because its terms keep escaping to very large positive values, very large negative values, or both.

The formal way to say this uses a bound M. For every real number M, you can find an index N so that all terms after N satisfy |a_n| > M. In plain language, no matter how large a cutoff you choose, the sequence eventually goes beyond it. That is the core idea behind the epsilon-style wording used later in sequence limits, except here the statement is about size rather than closeness.

A simple example is a_n = n. The terms 1, 2, 3, 4, ... keep growing, so the sequence is unbounded and does not converge. Another example is a_n = 2^n, which grows even faster. On the other side, a sequence like a_n = (-1)^n n is also unbounded, even though the signs switch back and forth, because the absolute value still gets larger and larger.

It helps to separate “unbounded” from “diverges” in your head. Every unbounded sequence is divergent, but not every divergent sequence is unbounded. For instance, a_n = (-1)^n diverges because it keeps oscillating, but it stays trapped between -1 and 1, so it is bounded.

Why unbounded sequence matters in Calculus II

Unbounded sequences show up all through the sequences unit in Calculus II because they are one of the clearest ways a sequence can fail to have a limit. Once you can spot unbounded growth, you can quickly decide that a sequence will not converge to a finite number, which saves time on limit questions and helps you classify behavior correctly.

This matters when you compare different formulas. Arithmetic sequences with a nonzero common difference can run off to infinity or negative infinity depending on the sign of the difference. Geometric sequences with |r| > 1 often become unbounded too, because repeated multiplication keeps increasing the magnitude of the terms. Seeing the pattern in the formula lets you predict the long-term behavior before you grind through many terms.

Unboundedness also gives you a useful contrast with bounded but divergent sequences. In Calculus II, that distinction comes up when you study whether a sequence has a limit, whether a series might behave nicely, and how a recurrence or explicit formula grows over time. When you know whether the terms stay trapped in an interval or keep escaping it, the rest of the analysis gets much easier.

How unbounded sequence connects across the course

Bounded Sequence

A bounded sequence stays between two fixed numbers, either above, below, or both. That is the opposite behavior from an unbounded sequence, which eventually leaves every finite interval. When you check a sequence, asking whether it is bounded is often the first step before deciding whether it can converge.

Convergent Sequence

A convergent sequence approaches a finite limit as n grows. An unbounded sequence cannot be convergent, because the terms do not stay near any one number. In practice, convergence questions often begin by checking whether the sequence even has a chance to stay bounded.

Divergent Sequence

Divergent sequence is the broader label for any sequence that does not converge. Unbounded sequences are one kind of divergent sequence, but divergence can also happen when the terms oscillate without settling down. That is why you check both the size of the terms and their pattern.

Limit of the Sequence

The limit of a sequence is the value the terms approach, if one exists. An unbounded sequence has no finite limit, so limit analysis often ends quickly once you show the terms grow without bound. This is especially common with sequences defined by powers, fractions, or recursive formulas.

Is unbounded sequence on the Calculus II exam?

A quiz or problem set may give you a formula like a_n = 3n - 1 or a_n = 2^n and ask whether the sequence converges, diverges, or is unbounded. The move is to look at the long-term growth of the terms, not just the first few values. If the expression grows past every finite number, you say it is unbounded and therefore not convergent.

You may also be asked to justify the claim with a limit argument. In that case, you connect the formula to growth behavior, such as terms increasing without bound or absolute values getting arbitrarily large. If the sequence alternates signs, check the magnitude too, because switching positive and negative does not make a sequence bounded.

Unbounded sequence vs Divergent Sequence

These are related but not the same. Unbounded means the terms escape every finite bound. Divergent only means the sequence does not converge, and a sequence can diverge while still staying bounded, like (-1)^n. If you see oscillation without growth, think divergent but bounded.

Key things to remember about unbounded sequence

• An unbounded sequence is one whose terms eventually exceed every finite bound in absolute value.

• If a sequence is unbounded, it cannot converge to a finite limit.

• Unbounded sequences can grow toward positive infinity, negative infinity, or switch signs while still growing in magnitude.

• A sequence can be divergent without being unbounded, so check both growth and oscillation.

• Arithmetic and geometric sequences often give quick examples of unbounded behavior.

Frequently asked questions about unbounded sequence

What is an unbounded sequence in Calculus II?

It is a sequence whose terms do not stay within any fixed finite range. As n gets large, the values keep getting larger in absolute value, so the sequence cannot settle to a finite limit. In Calculus II, this is one of the main ways to show a sequence does not converge.

Is every divergent sequence unbounded?

No. A sequence can diverge and still stay bounded. A classic example is (-1)^n, which bounces between -1 and 1 forever. Unbounded sequences are a smaller category inside divergent sequences.

How do you tell if a sequence is unbounded?

Look at the formula for growth. If the terms increase without limit, such as n, 2^n, or n^2, the sequence is unbounded. If the terms alternate signs, check the absolute value, because the magnitude can still grow even when the signs flip.

Can an unbounded sequence converge?

No. Convergence to a finite number requires the terms to get close to one fixed value. An unbounded sequence keeps leaving every finite range, so it cannot have a finite limit. That is why unboundedness is enough to rule out convergence right away.