If the terms do not approach zero, the series diverges. This is the fastest check and should be tried first.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
It only proves divergence
Terms approaching zero does not prove convergence. The test is inconclusive in that direction, always.
The harmonic series
The terms of 1 + 1/2 + 1/3 and so on do approach zero, yet the series diverges. It is the standard counterexample.
p-series
A p-series sums 1 ÷ nᵖ. It converges when p > 1 and diverges when p ≤ 1.
The boundary case
p = 1 is the harmonic series, which diverges. The boundary belongs to the divergent side.
Why they matter
Geometric and p-series are the comparison benchmarks. Most later tests work by comparing to one of them.
The first check
If the terms do not approach zero, the series diverges. It is the cheapest test and should be applied first. But if the terms do approach zero, the test is inconclusive and says nothing.
It can never prove convergence
Concluding convergence because the terms go to zero is the classic misuse. The harmonic series is the standing counterexample, and the exam relies on students remembering it.
The p-series family
The sum of 1/nᵖ converges when p > 1 and diverges when p ≤ 1. Memorising this gives a reference family for comparison tests, which is why it is singled out.
The harmonic series is the boundary case
With p = 1, the sum of 1/n diverges — slowly, but without bound. It sits exactly at the boundary of the p-series condition, which is why it is the most instructive example in the topic.
Step 2: Try It Yourself
Tap and try it out.
- Point(3, 0.33)
Step 3: Watch an Example
One step at a time.
Watch Marcus Apply the First Test
Marcus examines the series whose nth term is n ÷ (n + 1).
- Step 1
He takes the limit of the terms as n grows without bound.
Step 4: Your Turn
Practice makes it stick.
The Limit
Problem 1 of 2
The nth term is n ÷ (n + 1). What is its limit as n grows without bound?
The p-Series
Problem 2 of 2
The series summing 1 ÷ n². Does it converge? 1 yes, 0 no.
First Checks
1 of 8
p = 3. Does the p-series converge? 1 yes, 0 no.
2 of 8
p = 1. Does it converge? 1 yes, 0 no.
3 of 8
p = 0.5. Does it converge? 1 yes, 0 no.
4 of 8
The terms approach 0. Does the series converge? 1 yes, 0 no.
5 of 8
The nth term is 2n ÷ (n + 5). Limit as n grows?
6 of 8
That series therefore does what? 1 converges, 2 diverges.
7 of 8
Sort each p value by what the p-series does.
Tap something to move it.
- Empty
- Empty
8 of 8
p = 4. Does the p-series converge? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
p = 2.5. Does the p-series converge? 1 yes, 0 no.
Question 2 of 2
What can the nth term test establish?
What You Learned
- If the terms do not approach zero, the series diverges.
- Terms approaching zero proves nothing on its own.
- A p-series converges exactly when p > 1.