Formula Used
The nth term test studies this limit:
limn→∞ an
If lim an ≠ 0, then Σan diverges.
If lim an does not exist, then Σan diverges.
If lim an = 0, the test gives no final answer.
How to Use This Calculator
Select the closest sequence model. Enter its coefficient and related parameters. Use custom expression mode when your term does not match a listed form. Set sample values for numerical checking. Press calculate. The result appears above the form. Use CSV or PDF buttons to save the sampled work.
Example Data Table
| Series term |
Term limit |
Nth term verdict |
| 1 / n |
0 |
Inconclusive |
| n / (n + 1) |
1 |
Divergent |
| (-1)^n |
Does not exist |
Divergent |
| 1 / n^2 |
0 |
Inconclusive |
Understanding the Nth Term Test
The nth term test is a first screen for infinite series. It checks the behavior of the term sequence. It does not add the series. A series can only converge when its terms approach zero. If the terms approach a nonzero value, the series diverges. If the terms grow, the series also diverges. If the terms oscillate without settling at zero, the series diverges again.
Why This Test Matters
The test is useful because it is fast. Many series fail before harder tests are needed. For example, a series with terms n divided by n plus one has term limit one. Since one is not zero, the series diverges. A series with terms alternating between one and negative one also fails. Its limit does not exist.
Limits and Inconclusive Results
A zero limit does not prove convergence. It only means the series passes this first check. The harmonic series has terms one over n. Those terms approach zero, yet the series diverges. Another method is needed after a zero limit. Common choices include comparison tests, ratio tests, root tests, and integral tests.
Using Numerical Samples
This calculator samples large values of n. It also applies exact rules for common models. Numerical samples help reveal trends. They are not a full proof for every custom expression. Use them as guidance. For formal work, support the answer with algebraic limits.
Best Practice
Start with the simplest model. Enter the coefficient, power, ratio, or expression carefully. Increase the starting n when early terms are misleading. Use a smaller tolerance when values decay slowly. Read the verdict and explanation together. Remember the key rule. Nonzero or missing term limits prove divergence. Zero term limits require another test.
FAQs
What is the nth term test?
It is a divergence test for infinite series. It checks whether the sequence terms approach zero. If they do not approach zero, the series must diverge.
Can this test prove convergence?
Usually, no. If the term limit is zero, the result is inconclusive. You need another convergence test to make a final decision.
What happens when the limit is nonzero?
The series diverges. Infinite series cannot converge when their individual terms settle at a nonzero value.
What happens when the limit does not exist?
The series diverges. Missing, oscillating, or unbounded term limits fail the required condition for convergence.
Why is 1 divided by n inconclusive?
Its terms approach zero. The nth term test cannot decide more. In fact, the harmonic series diverges by other tests.
Can I enter a custom formula?
Yes. Use n as the variable. Functions like sin, cos, log, sqrt, abs, exp, and pow are supported.
What does tolerance mean?
Tolerance controls how close a sampled value must be to zero. It mainly helps classify custom numerical estimates.
Are numerical results always proof?
No. Numerical samples show likely behavior. For formal statistics or calculus work, confirm custom expressions with algebraic limit steps.