Nth Term Test- Divergence Test for Series
What Is the Nth Term Test (Divergence Test)?
The Nth Term Test, also called the Divergence Test, is the most basic test for checking whether an infinite series converges or diverges. If you're studying calculus or sequence convergence, this is where you start.
Here's the rule in plain English:
If the limit of the terms of a series does not equal zero, the series diverges.
That's it. That's the whole test. It won't tell you if a series converges—it only tells you when a series definitely diverges. If the limit equals zero, the test is inconclusive. You need other tests to figure out what happens next.
The Formal Definition
For a series ∑aₙ:
If lim(n→∞) aₙ ≠ 0, then the series diverges.
If lim(n→∞) aₙ = 0, the test is inconclusive.
Think of it this way: for a series to have any chance of converging, the individual terms must shrink toward zero. If they don't shrink to zero, they can't possibly add up to a finite number.
Why This Test Exists
You might wonder why mathematicians bother with a test that only catches divergences. The answer is simple: it's a quick elimination tool.
Before spending time on more complex tests like the Ratio Test or Integral Test, you check this first. If the terms don't approach zero, you're done—you know the series diverges. No further work needed.
Examples That Show Divergence
Example 1: Harmonic Series with Constant Terms
Consider ∑ 1. This is just 1 + 1 + 1 + 1 + ...
The Nth term is aₙ = 1.
lim(n→∞) 1 = 1 ≠ 0
The series diverges. This should be obvious—you're adding 1 infinitely many times.
Example 2: Series with Growing Terms
Consider ∑ n/(n+1).
The Nth term is aₙ = n/(n+1).
lim(n→∞) n/(n+1) = 1 ≠ 0
The series diverges. The terms approach 1, not zero.
Example 3: Alternating Constants
Consider ∑ (-1)ⁿ.
The Nth term oscillates between -1 and 1.
lim(n→∞) (-1)ⁿ does not exist.
The series diverges.
Examples Where the Test Is Inconclusive
Example 4: The Harmonic Series
Consider ∑ 1/n.
The Nth term is aₙ = 1/n.
lim(n→∞) 1/n = 0
The test is inconclusive. The terms approach zero, but the Nth Term Test doesn't tell you what happens next. Here's the bitter truth: the harmonic series diverges, even though its terms shrink to zero. You need the Integral Test or Cauchy Condensation Test to prove this.
Example 5: A Convergent P-Series
Consider ∑ 1/n².
The Nth term is aₙ = 1/n².
lim(n→∞) 1/n² = 0
The test is inconclusive here too. But unlike the harmonic series, this one converges. You'd use the P-Series Test to confirm it.
How to Apply the Nth Term Test: Step by Step
Here's the practical process for any series you encounter:
- Identify the Nth term (the formula for aₙ). Remove the summation notation and focus on the general term.
- Calculate the limit as n approaches infinity. Use whatever algebra you need—divide by the dominant term, apply L'Hôpital's Rule, or substitute large values.
- Check the result: If the limit ≠ 0 or doesn't exist → the series diverges. You're finished. If the limit = 0 → the test is inconclusive. Move on to another test.
Common Mistakes to Avoid
1. Assuming convergence when the limit equals zero.
Students often see lim aₙ = 0 and assume the series converges. This is wrong. The harmonic series is the classic counterexample. Terms approaching zero is necessary for convergence, but it's not sufficient.
2. Using the test on series that don't start at n=1.
The test still works for series starting at n=0 or any other index. Just compute the limit of aₙ as n→∞, regardless of where the series starts.
3. Forgetting that the limit must exist.
If the limit doesn't exist (oscillates, goes to infinity), the series diverges. A non-existent limit counts as "not equal to zero."
Nth Term Test vs. Other Convergence Tests
Here's how the Divergence Test compares to other tools in your toolkit:
| Test | What It Does | When to Use It |
|---|---|---|
| Nth Term Test | Checks if terms approach zero | First step for any series |
| Geometric Series Test | Checks if |r| < 1 | When you spot a common ratio |
| P-Series Test | Checks if p > 1 | When terms are 1/np |
| Ratio Test | Compares term ratios | Factorials and exponentials |
| Root Test | Checks nth roots | Powers of n in the base |
| Integral Test | Compares to improper integral | When terms are positive and decreasing |
Worked Example: Full Process
Determine whether ∑ (3n² + 2)/(5n² + 1) converges or diverges.
Step 1: Identify the Nth term.
aₙ = (3n² + 2)/(5n² + 1)
Step 2: Calculate the limit.
lim(n→∞) (3n² + 2)/(5n² + 1)
Divide numerator and denominator by n²:
= lim (3 + 2/n²)/(5 + 1/n²)
= 3/5
Step 3: Interpret the result.
lim aₙ = 3/5 ≠ 0
The series diverges. No further testing needed.
When to Skip the Nth Term Test
Some series make the test pointless from the start:
- Geometric series: If you spot a common ratio, use the geometric test directly. If |r| ≥ 1, it diverges anyway. If |r| < 1, it converges.
- Alternating series: Check the Alternating Series Test instead—it gives you actual convergence criteria.
- Telescoping series: Write out partial sums. The divergence test won't help here.
The Bottom Line
The Nth Term Test is a filter, not a complete solution. It eliminates series that clearly diverge. It does nothing for series that might converge.
Use it first, always. If the terms don't approach zero, you're done. If they do approach zero, move to the next test in your toolkit.
Master this test and the ones in the comparison table above, and you'll handle most convergence problems you're likely to encounter.