A series diverges whenever the sum of its terms fails to settle on a finite number. More precisely, if you keep a running total of the terms (called partial sums), and that running total grows without bound, oscillates, or otherwise never approaches a single value, the series diverges. Understanding exactly when this happens is one of the core skills in a calculus II course, and there are several reliable tests to check for it.
Partial Sums: The Core Idea
Every infinite series is really a sequence of partial sums. The first partial sum is just the first term. The second partial sum adds the first two terms. The third adds the first three, and so on. If that sequence of partial sums approaches a specific finite number as you include more and more terms, the series converges. If it doesn’t, for any reason, the series diverges.
Divergence can look different depending on the series. Sometimes the partial sums march off toward infinity (like 1 + 2 + 3 + 4 + …). Sometimes they swing back and forth without settling down (like 1 − 1 + 1 − 1 + …). Both behaviors count as divergence.
The Divergence Test (nth Term Test)
This is the quickest check and the one you should always try first. Look at the individual terms of the series, a_n, and take the limit as n goes to infinity. If that limit is anything other than zero, the series diverges. Period. The logic is straightforward: if the terms you’re adding don’t shrink to zero, the running total can never stabilize.
The critical trap here is the reverse. If the limit of a_n does equal zero, that tells you nothing by itself. The series might converge or it might not. The harmonic series (1 + 1/2 + 1/3 + 1/4 + …) is the classic example: every term gets smaller and approaches zero, yet the series still diverges. So the divergence test can confirm divergence but can never confirm convergence.
Why the Harmonic Series Diverges
The harmonic series is worth understanding on its own because it’s the most famous example of a series whose terms go to zero yet still diverges. The standard proof, dating back centuries, groups consecutive terms together and shows that each group adds more than 1/2 to the total. The first term is 1. The next term (1/2) is 1/2. The next two terms (1/3 + 1/4) sum to more than 1/2. The next four terms (1/5 through 1/8) also sum to more than 1/2. Each new group doubles in size but still exceeds 1/2, so the partial sums grow without limit. Slowly, yes, but they never stop growing.
Geometric Series
A geometric series has the form a + ar + ar² + ar³ + …, where r is the common ratio between consecutive terms. The rule is clean: the series diverges when the absolute value of r is greater than or equal to 1. When |r| is less than 1, each successive term shrinks fast enough for the sum to settle on a finite value. When |r| equals or exceeds 1, the terms either stay the same size or grow, making convergence impossible.
For example, 1 + 2 + 4 + 8 + … (r = 2) diverges because each term doubles. The series 1 + 1 + 1 + 1 + … (r = 1) diverges because the partial sums increase by 1 forever. And 1 − 1 + 1 − 1 + … (r = −1) diverges because the partial sums bounce between 0 and 1 without settling.
p-Series
A p-series takes the form 1/1^p + 1/2^p + 1/3^p + 1/4^p + …, where p is a positive real number. These series diverge when p is less than or equal to 1 and converge when p is greater than 1. The harmonic series is the p-series with p = 1, sitting right on the boundary. Anything with a smaller exponent (like 1/√n, where p = 1/2) diverges even faster. Anything with a larger exponent (like 1/n², where p = 2) converges.
This threshold is worth memorizing because p-series show up constantly as comparison benchmarks when you’re testing other series.
The Ratio and Root Tests
Both the ratio test and the root test give you a single number to evaluate, often called ρ (rho) or L. For the ratio test, you take the limit of |a_(n+1) / a_n| as n goes to infinity. For the root test, you take the limit of the nth root of |a_n|. The interpretation is identical for both:
- ρ > 1 (or ρ = ∞): the series diverges.
- ρ < 1: the series converges.
- ρ = 1: the test is inconclusive, and you need a different method.
The ratio test works especially well for series involving factorials or exponentials, where consecutive terms have a clean ratio. The root test shines when the entire term is raised to the nth power. When ρ lands exactly on 1, neither test can help, which happens more often than you’d like (the harmonic series, for instance, gives ρ = 1 for both tests).
The Integral Test
If you can write the terms of your series as a function f(x) that is continuous, positive, and decreasing from some point onward, the integral test links the series directly to an improper integral. If the integral of f(x) from some starting value to infinity diverges, so does the series. If the integral converges, the series converges too.
This is one of the cleanest connections in calculus: the series and its corresponding improper integral share the same convergence behavior. You’re essentially asking whether the area under a curve is finite, which for many functions is easier to evaluate than summing terms directly. One important note: even when the integral test confirms convergence, it does not tell you the actual value of the series. It only answers the yes-or-no question.
Comparison and Limit Comparison Tests
Sometimes the best way to determine divergence is to compare your series to one you already know diverges. The direct comparison test says that if every term of your series is at least as large as the corresponding term of a known divergent series (with all positive terms), your series diverges too. Bigger than a divergent series means divergent.
The limit comparison test is more flexible. Take two series with positive terms and compute the limit of a_n / b_n as n goes to infinity. If that limit is a positive finite number, both series behave the same way: they either both converge or both diverge. If the limit is infinity and the series in the denominator diverges, the series in the numerator diverges as well, since its terms are growing faster than those of an already-divergent series.
In practice, you’ll often compare an unfamiliar series to a p-series or geometric series because you already know exactly when those diverge.
Alternating Series
An alternating series has terms that flip between positive and negative (like 1 − 1/2 + 1/3 − 1/4 + …). The alternating series test can confirm convergence when the absolute values of the terms decrease steadily to zero, but it cannot directly confirm divergence. If the terms don’t decrease to zero, the test simply doesn’t apply.
To show an alternating series diverges, you typically fall back on the divergence test. Check whether the limit of the full term (including the sign changes) exists and equals zero. If that limit doesn’t exist or isn’t zero, the series diverges. For instance, the series whose terms alternate as (−1)^n · n / (n + 1) diverges because the terms approach (−1)^n · 1, bouncing between values near 1 and −1 rather than shrinking to zero.
Telescoping Series
A telescoping series is one where most terms cancel when you write out the partial sums, leaving only a few terms from the beginning and end. Whether it converges or diverges comes down to what happens to those surviving terms. You compute the partial sum, cancel everything that cancels, and take the limit of what remains. If that limit is finite, the series converges. If the remaining terms grow without bound or fail to approach a fixed value, the series diverges.
There’s no special shortcut here. Telescoping series are diagnosed by directly examining their partial sums, which is really just the fundamental definition of divergence applied in the most literal way possible.
Choosing the Right Test
With so many tests available, picking the right one comes down to pattern recognition. If the terms clearly don’t approach zero, use the divergence test and you’re done. If the series is geometric, check whether |r| ≥ 1. If it’s a p-series, check whether p ≤ 1. For series with factorials or exponential terms, the ratio test is usually the cleanest path. For series that look like a known divergent form but are slightly messier, comparison or limit comparison will often work. And when you can easily integrate the corresponding function, the integral test gives a definitive answer.
No single test handles every case, and some series require trying more than one approach before you find one that gives a clear result. The ratio and root tests, in particular, go inconclusive (ρ = 1) on many common series, which is not a failure of your approach but a signal to switch tools.