Divergent Series

Q: Can a divergent series have terms that approach zero?

Yes. The most famous example is the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$. Each term $\frac{1}{n}$ approaches 0, yet the partial sums grow without bound, so the series diverges. Having terms that shrink to zero is necessary for convergence but not sufficient.

Divergent Series

A series that does not converge. For example, the series 1 + 2 + 3 + 4 + 5 + ··· diverges. Its sequence of partial sums 1, 1 + 2, 1 + 2 + 3 , 1 + 2 + 3 + 4 , 1 + 2 + 3 + 4 + 5, ... diverges.

Key Formula

\text{A series } \sum_{n=1}^{\infty} a_n \text{ is divergent if } \lim_{N \to \infty} S_N \text{ does not exist (or is } \pm\infty\text{)}

Where:

$a_n$ = The nth term of the series
$S_N$ = The Nth partial sum, equal to $a_1 + a_2 + \cdots + a_N$
$N$ = The number of terms included in the partial sum

Worked Example

Problem: Determine whether the series 1 + 2 + 3 + 4 + ··· diverges by examining its partial sums.

Step 1: Write out the first several partial sums. Each partial sum adds one more term to the running total.

S_1 = 1,\quad S_2 = 1+2 = 3,\quad S_3 = 1+2+3 = 6,\quad S_4 = 1+2+3+4 = 10

Step 2: Use the formula for the sum of the first N positive integers to express the general partial sum.

S_N = \frac{N(N+1)}{2}

Step 3: Evaluate the limit of the partial sums as N grows without bound.

\lim_{N \to \infty} \frac{N(N+1)}{2} = \infty

Step 4: Because the partial sums grow without bound, the limit does not exist as a finite number. The series therefore diverges.

Answer: The series 1 + 2 + 3 + 4 + ··· is divergent because its partial sums grow to infinity.

Another Example

Problem: Show that the series 1 − 1 + 1 − 1 + 1 − 1 + ··· (Grandi's series) is divergent.

Step 1: Write the series in summation notation and compute the first several partial sums.

\sum_{n=0}^{\infty} (-1)^n: \quad S_1 = 1,\quad S_2 = 1 - 1 = 0,\quad S_3 = 1,\quad S_4 = 0

Step 2: The partial sums alternate between 1 and 0 forever. They do not approach any single value.

S_N = \begin{cases} 1 & \text{if } N \text{ is odd} \\ 0 & \text{if } N \text{ is even} \end{cases}

Step 3: Since the limit of the partial sums does not exist, the series diverges — even though the partial sums stay bounded.

Answer: Grandi's series diverges by oscillation. The partial sums bounce between 0 and 1 and never settle on a single limit.

Frequently Asked Questions

Can a divergent series have terms that approach zero?

Yes. The most famous example is the harmonic series

\sum_{n=1}^{\infty} \frac{1}{n}

. Each term

\frac{1}{n}

approaches 0, yet the partial sums grow without bound, so the series diverges. Having terms that shrink to zero is necessary for convergence but not sufficient.

How do you quickly tell if a series diverges?

The simplest check is the Divergence Test (also called the nth-term test): if

\lim_{n \to \infty} a_n \neq 0

, the series

\sum a_n

must diverge. However, if the limit does equal 0, the test is inconclusive, and you need other methods like the ratio test, comparison test, or integral test.

Divergent Series vs. Convergent Series

A convergent series has partial sums that approach a specific finite number — that number is called the sum of the series. A divergent series, by contrast, has partial sums that fail to approach any finite limit; they may increase without bound, decrease without bound, or oscillate. Every infinite series is one or the other: convergent or divergent.

Why It Matters

Recognizing whether a series converges or diverges is essential throughout calculus, physics, and engineering. Many real-world quantities — such as signal processing coefficients, probability distributions, and power series representations of functions — are expressed as infinite sums, and using a divergent series as though it has a finite value leads to incorrect results. Understanding divergence also motivates key convergence tests that you will use repeatedly in later mathematics courses.

Common Mistakes

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

Correction: Terms approaching zero is necessary but not sufficient. The harmonic series

\sum 1/n

is the classic counterexample: its terms go to zero, yet it diverges. Always apply a proper convergence test beyond the nth-term test.

Mistake: Confusing a divergent series with a divergent sequence.

Correction: A series involves summing terms (partial sums), while a sequence is just a list of individual values. A series can diverge even when the underlying sequence of terms converges (again, think of the harmonic series). Make sure you are analyzing the partial sums, not just the terms.

Related Terms

Series — General concept of summing infinitely many terms
Convergent Series — The opposite — series whose partial sums have a finite limit
Sequence of Partial Sums — The running totals used to test convergence
Diverge — The general concept of failing to approach a limit
Divergent Sequence — A sequence whose terms do not settle to a limit
Converge — The general concept of approaching a finite limit
Convergent Sequence — A sequence whose terms approach a specific value