Convergent Sequence

Convergent Sequence

A sequence with a limit that is a real number. For example, the sequence 2.1, 2.01, 2.001, 2.0001, . . . has limit 2, so the sequence converges to 2. On the other hand, the sequence 1, 2, 3, 4, 5, 6, . . . has a limit of infinity (∞). This is not a real number, so the sequence does not converge. It is a divergent sequence.

Key Formula

\lim_{n \to \infty} a_n = L

Where:

$a_n$ = The nth term of the sequence
$n$ = The index (position) of the term, starting from 1 or 0
$L$ = The limit — a specific real number the terms approach

Worked Example

Problem: Determine whether the sequence defined by a_n = 1/n converges, and if so, find its limit.

Step 1: Write out the first several terms of the sequence to see the pattern.

a_1 = 1,\quad a_2 = \frac{1}{2},\quad a_3 = \frac{1}{3},\quad a_4 = \frac{1}{4},\quad a_{10} = \frac{1}{10},\quad a_{100} = \frac{1}{100}

Step 2: Observe what happens as n gets very large. As n increases, 1/n gets closer and closer to 0.

a_{1000} = \frac{1}{1000} = 0.001, \quad a_{1{,}000{,}000} = 0.000001

Step 3: Evaluate the limit formally. Since the numerator stays fixed at 1 while the denominator grows without bound, the fraction shrinks toward 0.

\lim_{n \to \infty} \frac{1}{n} = 0

Step 4: Since the limit equals 0, which is a real number, the sequence converges.

Answer: The sequence a_n = 1/n converges, and its limit is 0.

Another Example

Problem: Determine whether the sequence defined by a_n = (-1)^n converges.

Step 1: Write out the first several terms.

a_1 = -1,\quad a_2 = 1,\quad a_3 = -1,\quad a_4 = 1,\quad a_5 = -1, \ldots

Step 2: The terms alternate between −1 and 1 forever. They do not settle down toward any single real number.

Step 3: Because no single real number L satisfies the condition that terms eventually stay arbitrarily close to L, the limit does not exist.

\lim_{n \to \infty} (-1)^n \text{ does not exist}

Answer: The sequence a_n = (-1)^n does not converge. It is a divergent sequence because it oscillates and never approaches a single value.

Frequently Asked Questions

How do you tell if a sequence converges or diverges?

Compute the limit of the general term a_n as n approaches infinity. If that limit equals a specific real number, the sequence converges. If the limit is infinity, negative infinity, or does not exist (for example, the terms oscillate), the sequence diverges.

Can a convergent sequence have terms that are never exactly equal to its limit?

Yes. The sequence 1/n converges to 0, but no term in the sequence is actually 0. Convergence only requires that the terms get arbitrarily close to the limit, not that they ever reach it.

Convergent Sequence vs. Divergent Sequence

A convergent sequence has terms that approach a specific real number as the index grows. A divergent sequence does not — its terms may grow without bound (like 1, 2, 3, …), oscillate (like −1, 1, −1, 1, …), or behave erratically. Every sequence is either convergent or divergent; there is no middle ground.

Why It Matters

Convergent sequences are fundamental in calculus and analysis because they formalize the idea of a process approaching a definite result. Many key concepts — limits of functions, derivatives, integrals, and infinite series — are all built on the notion of convergence. In applied fields, convergent sequences describe systems that stabilize, such as iterative algorithms that zero in on a solution.

Common Mistakes

Mistake: Thinking a sequence that "goes to infinity" converges because it has a predictable pattern.

Correction: Infinity is not a real number. A sequence like 1, 2, 3, 4, … grows without bound, so it diverges even though it follows a clear rule.

Mistake: Confusing a convergent sequence with a convergent series.

Correction: A convergent sequence concerns the terms a_n approaching a limit. A convergent series concerns the sum a_1 + a_2 + a_3 + … approaching a finite value. A sequence can converge (e.g., 1/n → 0) while the corresponding series diverges (the harmonic series 1 + 1/2 + 1/3 + … = ∞).

Related Terms

Sequence — Ordered list of numbers a convergent sequence is
Limit — The value a convergent sequence approaches
Divergent Sequence — A sequence that does not converge
Convergent Series — Infinite sum that approaches a finite value
Divergent Series — Infinite sum that does not have a finite value
Converge — General term for approaching a definite value
Real Numbers — The limit must be a real number
Infinity — Not a real number, so sequences tending to ∞ diverge