Divergence Test Calculator

Calculator Inputs

Pick a supported sequence form. Then enter values and generate the divergence test result.

Sequence model

Sample terms to generate

Coefficient A

Numerator power p

Denominator power q

Numerator constant B

Denominator constant C

Base r

Exponent multiplier k

Log power m

Log shift s

Known limit type

Known finite limit value

Example Data Table

Series	Term aₙ	Term Limit	Divergence Test Result
∑ 1/n	1/n	0	Inconclusive
∑ n/(n+1)	n/(n+1)	1	Diverges
∑ 2ⁿ	2ⁿ	∞	Diverges
∑ ln(n)/n	ln(n)/n	0	Inconclusive
∑ (-1)ⁿ	(-1)ⁿ	Does not exist	Diverges

Formula Used

The divergence test examines the limit of the sequence term aₙ in the infinite series ∑aₙ.

Main test: If lim aₙ ≠ 0, or the limit does not exist, then ∑aₙ diverges.

Important note: If lim aₙ = 0, the test is inconclusive. Another convergence test is needed.

Supported sequence forms

Power ratio: aₙ = A·n^p / n^q
Shifted rational: aₙ = A·(n^p + B) / (n^q + C)
Exponential: aₙ = A·r^(k·n)
Log quotient: aₙ = A·[ln(n + s)]^m / n^q
Known limit mode: use this when you already know lim aₙ

The calculator uses asymptotic growth comparisons for the selected model. It then converts that term limit into a divergence test decision.

How to Use This Calculator

Select the sequence model closest to your series term aₙ.
Enter the needed coefficients, powers, constants, or base values.
Choose how many sample terms you want in the table and graph.
Click Run Divergence Test to generate the result.
Read the term limit and the test conclusion shown above the form.
Use CSV or PDF export if you want to save the output.
When the result says inconclusive, apply another convergence test next.

This tool is best for quick screening. It tells you when divergence is certain, but it cannot confirm convergence when the term limit equals zero.

FAQs

1. What does the divergence test check?

It checks the limit of the sequence term aₙ in a series ∑aₙ. If that limit is not zero, or does not exist, the series definitely diverges.

2. Does lim aₙ = 0 mean the series converges?

No. A zero term limit is only necessary, not sufficient. The harmonic series is a classic counterexample because its terms approach zero while the series still diverges.

3. When is this calculator most useful?

It is useful as a first check for infinite series. It quickly identifies obvious divergence before you try stronger tests like comparison, ratio, root, or integral methods.

4. Why does a nonzero term limit force divergence?

A convergent series must have terms shrinking to zero. If the terms stay away from zero, or keep oscillating without settling, partial sums cannot stabilize.

5. Can I use this for alternating series?

Yes, but only for the divergence test itself. If the term limit fails to exist or is not zero, divergence follows. If the limit is zero, use another alternating-series method.

6. What if my term formula is not listed?

Use known limit mode when you already know lim aₙ. Otherwise, rewrite the term into a similar growth form or use symbolic work before applying the calculator.

7. Why do exported values sometimes look approximate?

The table and chart display numeric samples of the sequence. They support interpretation, but the actual decision comes from the term-limit rule and growth comparison logic.

8. Does the graph prove convergence or divergence?

No. The graph is visual help only. It shows sample terms, while the divergence decision comes from the mathematical limit of aₙ as n grows large.