Nth Term Test Calculator
Infinite series can be difficult to evaluate, especially when determining whether they converge or diverge. One of the simplest and most commonly used tools in calculus for this purpose is the Nth Term Test for Divergence.
Our Nth Term Test Calculator is a tool designed to help you quickly determine if an infinite series diverges based on its nth term formula. Simply input your formula, and this calculator will evaluate the limit as n approaches infinity and determine whether the series diverges.
Formula
The Nth Term Test for Divergence (also called the Test for Divergence) says:
If the limit of the nth term of a series does not equal zero, then the series diverges.
Mathematically:
- If
lim (n→∞) aₙ ≠ 0, then the sum ∑ aₙ diverges. - If
lim (n→∞) aₙ = 0, the test is inconclusive — the series may or may not converge.
So, the test only tells us if a series diverges. If the nth term goes to zero, you must use other tests (like Ratio, Root, Integral, etc.) to determine convergence.
How to Use the Nth Term Test Calculator
Steps:
- Enter the formula for the nth term of your series.
Usenas the variable. Examples:1/n,n/(n+1),1/n^2, etc. - Click “Calculate”.
- The calculator evaluates the limit of aₙ as
n → ∞. - Based on the result:
- If the limit ≠ 0 → the series diverges
- If the limit = 0 → test is inconclusive
Example
Series:
∑ 1/n
Step-by-step:
- aₙ = 1/n
- lim (n→∞) 1/n = 0
- Result: The Nth Term Test is inconclusive
(However, we know from other tests that this diverges — it’s the harmonic series)
FAQs
1. What is the Nth Term Test?
It checks whether the terms of a series go to zero. If they don’t, the series diverges.
2. What’s the rule?
If lim (n→∞) aₙ ≠ 0, the series diverges. If it equals 0, you need more tests.
3. What types of series is this test good for?
All infinite series. It’s a quick way to rule out convergence.
4. What if the limit is undefined or infinite?
Then the series diverges.
5. What if the limit is zero?
Then the test is inconclusive — it tells you nothing about convergence.
6. Can I input functions with powers or factorials?
Yes! You can use expressions like 1/n^2, n/(n+1), or even 1/n!.
7. Does this calculator show the exact limit?
It numerically approximates the limit using very large values of n.
8. Is this calculator accurate for all types of series?
It works for expressions that return real numbers for large n. It may not handle oscillating or symbolic inputs well.
9. What if I input a series like (–1)ⁿ/n?
It will return the numerical result — for that case, it converges but the test is inconclusive.
10. Is this calculator mobile-friendly?
Yes, it’s designed to work on all devices.
11. Can I input trigonometric or exponential functions?
You can use simple JS functions, but for complex expressions, it’s best to rewrite them using algebraic notation (e.g., Math.sin(n) is not supported here).
12. What happens if I input something invalid?
You’ll get a syntax error prompt.
13. Can I use this in calculus exams?
Yes — it’s a great study companion, but not a replacement for understanding the concepts.
14. Is the calculator fast?
Yes. It uses JavaScript to instantly evaluate and display results.
15. Can it replace other convergence tests?
No. It only detects divergence — for convergence, use Ratio, Root, Integral, Comparison, etc.
16. Is it free to use?
Yes. No downloads, no fees.
17. Can I embed this on my website?
Absolutely. Ask and we’ll give you the embeddable code.
18. Is this calculator safe?
Yes. It runs entirely in your browser, without storing or sending data.
19. What’s the minimum value of n used in the calculation?
It starts evaluating near n = 1,000,000 to approximate the limit.
20. Can this be used for sequences instead of series?
Yes. The limit evaluated here is the same used to check if a sequence converges.
Conclusion
The Nth Term Test Calculator is a quick and useful way to evaluate whether an infinite series diverges. While it doesn’t guarantee convergence when the limit equals zero, it definitively rules out convergence when the limit isn’t zero.