Nth Term Test Calculator

Enter nth Term Expression (use ‘n’ as variable):

Calculate

Determining whether an infinite series converges or diverges is a core part of calculus. The Nth Term Test, also called the Divergence Test, is one of the simplest methods for checking divergence. This test evaluates the limit of the general term ana_nan​ as n→∞n \to \inftyn→∞. If this limit is not zero, the series diverges.

Our Nth Term Test Calculator automates this process, allowing students and professionals to quickly determine whether a series converges or diverges based on the behavior of its terms.

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Formula
To use the Nth Term Test, evaluate the following:

If lim⁡n→∞an≠0\lim_{n \to \infty} a_n \neq 0n→∞lim​an​=0

then the series ∑an\sum a_n∑an​

diverges.

If lim⁡n→∞an=0\lim_{n \to \infty} a_n = 0n→∞lim​an​=0

the test is inconclusive—the series may or may not converge.

This is a necessary condition for convergence, but not sufficient.

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How to Use

1. Enter the nth term expression using n as the variable (e.g., 1/n, (n+1)/(2n+3)).
2. Click “Calculate” — The tool computes the limit of the term as n→∞n \to \inftyn→∞.
3. Interpret the Result — If the limit is nonzero, the series diverges. If it’s zero, the test is inconclusive.

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Example

Input:
Expression = 1/n

Calculation: lim⁡n→∞1n=0\lim_{n \to \infty} \frac{1}{n} = 0n→∞lim​n1​=0

Result:
The limit is 0, so the test is inconclusive. You must apply another convergence test.

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FAQs

1. What is the Nth Term Test?
   It checks whether a series diverges by evaluating the limit of its general term.
2. What if the limit is not 0?
   Then the series definitely diverges.
3. What if the limit is 0?
   The test is inconclusive—you need to try another method (like the ratio or integral test).
4. Can this test prove convergence?
   No, it can only prove divergence.
5. Is this calculator accurate for all functions?
   It supports most algebraic and rational expressions using n.
6. What if I enter a constant (e.g., 1)?
   Limit is 1 → the series diverges.
7. Does this test apply to finite series?
   No—it’s designed for infinite series only.
8. Can I enter factorials or exponents?
   Yes! Use n!, 2^n, n^2, etc.
9. Does it support trigonometric functions?
   Yes, you can enter expressions like sin(n)/n.
10. What does “inconclusive” mean?
    It means you can’t decide convergence from this test alone.
11. What’s a good next step after an inconclusive result?
    Try Ratio, Root, Comparison, or Integral Tests.
12. Does this handle limits at other points?
    No, only as n→∞n \to \inftyn→∞.
13. Why is the test important?
    It’s a quick first step in checking series divergence.
14. What series always fail this test?
    Any whose nth term doesn’t approach 0 (e.g., 1, n/(n+1), cos(n)).
15. Can I use this in class or homework?
    Yes—it’s perfect for learning and validation.
16. Is this a rigorous proof tool?
    No, it’s a calculator for quick analysis, not formal proof.
17. Does this use L’Hôpital’s Rule?
    Internally, yes—Math.js may apply such techniques for limits.
18. Is it safe to enter complicated expressions?
    Yes, but make sure syntax is correct (use parentheses!).
19. Is this calculator mobile-friendly?
    Yes, it works on smartphones and tablets.
20. Is it free?
    Absolutely—use it anytime for learning or work.

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Conclusion
The Nth Term Test Calculator is a fast and reliable way to determine whether an infinite series diverges. It’s perfect for students studying calculus, teachers explaining convergence concepts, and professionals validating mathematical series.