Convergent Series

Q: How do you tell if a series converges or diverges?

You examine whether the sequence of partial sums approaches a finite limit. Several tests can help: the geometric series test checks if |r| < 1, the ratio test compares consecutive terms, and the integral test uses a related function. A necessary (but not sufficient) first check is that the individual terms must approach zero; if they don't, the series definitely diverges.

Q: Does a series converge if its terms go to zero?

Not necessarily. The terms approaching zero is required for convergence, but it does not guarantee it. The classic counterexample is the harmonic series 1 + 1/2 + 1/3 + 1/4 + ···, where the terms go to zero yet the partial sums grow without bound, so the series diverges.

Convergent Series

An infinite series for which the sequence of partial sums converges. For example, the sequence of partial sums of the series 0.9 + 0.09 + 0.009 + 0.0009 + ··· is 0.9, 0.99, 0.999, 0.9999, .... This sequence converges to 1, so the series 0.9 + 0.09 + 0.009 + 0.0009 + ··· is convergent.

See also

Sequence, divergent series

Key Formula

\text{A series } \sum_{n=1}^{\infty} a_n \text{ converges if } \lim_{N \to \infty} S_N = \lim_{N \to \infty} \sum_{n=1}^{N} a_n = S

Where:

$a_n$ = The nth term of the series
$S_N$ = The Nth partial sum, equal to the sum of the first N terms
$S$ = The finite number the partial sums approach (the sum of the series)

Worked Example

Problem: Determine whether the geometric series 6 + 3 + 3/2 + 3/4 + ··· converges, and if so, find its sum.

Step 1: Identify the first term and the common ratio. Each term is half the previous term.

a = 6, \quad r = \frac{3}{6} = \frac{1}{2}

Step 2: Check the convergence condition for a geometric series: the series converges when the absolute value of the common ratio is less than 1.

|r| = \left|\frac{1}{2}\right| = \frac{1}{2} < 1 \quad \checkmark

Step 3: Write out the first few partial sums to see them approaching a limit.

S_1 = 6, \quad S_2 = 9, \quad S_3 = 10.5, \quad S_4 = 11.25

Step 4: Apply the geometric series sum formula to find the exact value the partial sums approach.

S = \frac{a}{1 - r} = \frac{6}{1 - \frac{1}{2}} = \frac{6}{\frac{1}{2}} = 12

Answer: The series converges, and its sum is 12.

Another Example

Problem: Show that the series 1 + 1/2 + 1/4 + 1/8 + ··· converges and find its sum.

Step 1: Identify the first term and common ratio.

a = 1, \quad r = \frac{1}{2}

Step 2: Since |r| = 1/2 < 1, the geometric series converges. Compute partial sums to build intuition.

S_1 = 1, \quad S_2 = 1.5, \quad S_3 = 1.75, \quad S_4 = 1.875

Step 3: Use the formula to find the exact sum.

S = \frac{1}{1 - \frac{1}{2}} = 2

Answer: The series converges to 2. Notice the partial sums get closer and closer to 2 without ever exceeding it.

Frequently Asked Questions

How do you tell if a series converges or diverges?

You examine whether the sequence of partial sums approaches a finite limit. Several tests can help: the geometric series test checks if |r| < 1, the ratio test compares consecutive terms, and the integral test uses a related function. A necessary (but not sufficient) first check is that the individual terms must approach zero; if they don't, the series definitely diverges.

Does a series converge if its terms go to zero?

Not necessarily. The terms approaching zero is required for convergence, but it does not guarantee it. The classic counterexample is the harmonic series 1 + 1/2 + 1/3 + 1/4 + ···, where the terms go to zero yet the partial sums grow without bound, so the series diverges.

Convergent Series vs. Divergent Series

A convergent series has partial sums that settle toward a finite number. A divergent series does not — its partial sums may grow without bound (like 1 + 2 + 3 + ···), oscillate (like 1 − 1 + 1 − 1 + ···), or otherwise fail to approach a single finite value. Every infinite series is one or the other: it either converges or diverges.

Why It Matters

Convergent series appear throughout science and engineering whenever you approximate a quantity with an infinite process. For example, Taylor series let you represent functions like

\sin x

and

e^x

as convergent infinite sums, which is how calculators evaluate these functions. Understanding convergence also matters in finance (computing the present value of perpetual cash flows) and physics (summing contributions from infinitely many sources).

Common Mistakes

Mistake: Assuming a series converges just because its terms approach zero.

Correction: Terms going to zero is necessary but not sufficient. The harmonic series 1 + 1/2 + 1/3 + 1/4 + ··· has terms going to zero yet diverges. You must verify convergence with a proper test.

Mistake: Confusing a convergent sequence with a convergent series.

Correction: A convergent sequence is a list of numbers approaching a limit. A convergent series is an infinite sum whose partial sums approach a limit. A sequence of terms can converge (to zero, for instance) while the series formed by adding those terms diverges.

Related Terms

Infinite Series — The broader category that includes convergent series
Sequence of Partial Sums — The tool used to define series convergence
Divergent Series — A series that does not converge
Convergent Sequence — A sequence whose terms approach a finite limit
Converge — The general concept of approaching a limit
Series — The sum of the terms of a sequence
Sequence — An ordered list of numbers underlying a series