The Nth Term Test Calculator is a powerful online tool designed to help students, mathematicians, and researchers determine whether an infinite series converges or diverges by examining the behavior of its terms as nnn approaches infinity. This test, also known as the Test for Divergence, is a fundamental concept in calculus and real analysis, forming part of the toolkit for studying infinite series.
Nth Term Test Calculator (Divergence Test)
How to Use the Nth Term Test Calculator
Using the Nth Term Test Calculator is straightforward. Follow these steps to get precise results:
- Enter the general term ana_nan – This is the formula for the terms in the series. For example, an=1na_n = \frac{1}{n}an=n1 or an=nn+1a_n = \frac{n}{n+1}an=n+1n.
- Specify the variable – Usually, this will be nnn, representing the term number in the sequence.
- Click the “Calculate” button – The tool computes the limit of ana_nan as nnn approaches infinity.
- Review the result – The calculator will tell you whether the series diverges (if the limit is not zero) or if the test is inconclusive (limit is zero).
Formula for the Nth Term Test
The mathematical basis for the calculator is simple but powerful.
Given a series:
S = a₁ + a₂ + a₃ + … + aₙ + …
We look at the nth term ana_nan.
The Nth Term Test states:
- If lim (n → ∞) aₙ ≠ 0, the series diverges.
- If lim (n → ∞) aₙ = 0, the test is inconclusive (you need another test to determine convergence).
In plain text:
- Find the limit of aₙ as n approaches infinity.
- If the limit is not zero (including infinity), the series diverges.
- If the limit is zero, the test cannot confirm convergence — further analysis is required.
Example Calculations
Example 1: Divergent Series
Series: ∑n=1∞nn+1\sum_{n=1}^{\infty} \frac{n}{n+1}∑n=1∞n+1n
Step 1: General term is an=nn+1a_n = \frac{n}{n+1}an=n+1n.
Step 2: Find the limit:
lim (n → ∞) nn+1=1\frac{n}{n+1} = 1n+1n=1.
Step 3: Since the limit ≠ 0, the series diverges.
Result: Divergent series.
Example 2: Inconclusive Case
Series: ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}∑n=1∞n1
Step 1: General term is an=1na_n = \frac{1}{n}an=n1.
Step 2: Find the limit:
lim (n → ∞) 1n=0\frac{1}{n} = 0n1=0.
Step 3: Since the limit = 0, the Nth Term Test is inconclusive. (We know from the Harmonic Series test that it diverges, but the Nth Term Test alone does not tell us this.)
Result: Inconclusive — use another test.
Example 3: Convergent Series (Requires Additional Test)
Series: ∑n=1∞1n2\sum_{n=1}^{\infty} \frac{1}{n^2}∑n=1∞n21
Step 1: General term is an=1n2a_n = \frac{1}{n^2}an=n21.
Step 2: Find the limit:
lim (n → ∞) 1n2=0\frac{1}{n^2} = 0n21=0.
Step 3: Limit is zero, so the test is inconclusive — but the p-series test shows convergence.
Result: Inconclusive — other tests needed.
Additional Insights
- The Nth Term Test only detects divergence, not convergence.
- If the terms of the series do not approach zero, the sum cannot exist in the finite sense.
- This test is often the first check before attempting more advanced methods.
- It applies to both positive term series and alternating series.
- Be careful with limits involving oscillations (e.g., sin(n)) — they do not approach zero and thus diverge.
20 Frequently Asked Questions (FAQs)
1. What is the Nth Term Test?
It’s a test in calculus used to check if an infinite series diverges based on the limit of its terms.
2. Can the Nth Term Test prove convergence?
No, it can only prove divergence. If the limit is zero, more tests are needed.
3. When should I use the Nth Term Test?
Use it as the first step when analyzing an infinite series.
4. What if the limit of aₙ is not zero?
The series diverges.
5. What if the limit of aₙ is zero?
The test is inconclusive — try another convergence test.
6. Does this work for alternating series?
Yes, but you may also need the Alternating Series Test for full analysis.
7. Is it possible for a series with aₙ → 0 to diverge?
Yes, the harmonic series is a classic example.
8. What’s the difference between sequence and series in this context?
A sequence is a list of numbers; a series is the sum of those numbers.
9. Why is this test also called the Test for Divergence?
Because it directly identifies divergence when the nth term does not go to zero.
10. Do I need calculus to use the Nth Term Test?
You need basic limit calculation skills from calculus.
11. Can the test handle functions with oscillations?
Yes, if the terms fail to approach zero due to oscillation, it implies divergence.
12. What is the main limitation of the test?
It cannot confirm convergence.
13. How does the calculator handle infinite limits?
If the limit is infinity, it declares divergence.
14. Can this test be used on finite sums?
No, it’s specifically for infinite series.
15. What about p-series?
The test alone is inconclusive if p > 1 — you need the p-series test.
16. Can it be used for complex numbers?
Yes, if you check the absolute value of the terms.
17. Does the test require positive terms?
No, it works regardless of term sign.
18. How do I know when to stop using this test?
If the limit is zero, stop — use another test.
19. What happens if aₙ does not exist for some n?
The series may not be well-defined.
20. Is the Nth Term Test always the first step?
It’s highly recommended as the first screening step in series analysis.