Convergence Comparison Test Calculator

Comparison method

All terms are assumed positive for sufficiently large n.

Starting n

Ending n

Original series label

Benchmark series label

Model used

Term form: c / [n^p (ln n)^q gⁿ]

Set q = 0 to remove the logarithmic factor. Set g = 1 to remove the geometric factor.

Original coefficient c

Original power p of n

Original power q of ln n

Original geometric base g

Benchmark coefficient c

Benchmark power p of n

Benchmark power q of ln n

Benchmark geometric base g

Reset

Example data table

This sample uses limit comparison with aₙ = 4 / n² and bₙ = 1 / n².

n	aₙ	bₙ	aₙ / bₙ	Observation
5	0.16	0.04	4	Finite positive ratio
10	0.04	0.01	4	Same dominant order
20	0.01	0.0025	4	Both series converge

Formula used

aₙ = c₁ / [n^p₁ (ln n)^q₁ g₁^n]
bₙ = c₂ / [n^p₂ (ln n)^q₂ g₂^n]

Limit comparison idea: compute L = lim (aₙ / bₙ). If 0 < L < ∞, both series have the same behavior.

Direct comparison idea: if aₙ ≤ Kbₙ eventually and bₙ converges, then aₙ converges. If aₙ ≥ kbₙ eventually and bₙ diverges, then aₙ diverges.

Benchmark rules used here: if g > 1, the series converges. If g = 1, it converges when p > 1, or when p = 1 and q > 1. Otherwise it diverges.

How to use this calculator

Choose limit comparison or direct comparison.
Enter the original series parameters c, p, q, and g.
Enter a benchmark series with known behavior.
Select a practical n-range for the sample table and graph.
Click Analyze Series to view the result above the form.
Use the table and chart to inspect how the ratio behaves.
Export the computed rows as CSV or PDF.

FAQs

1) What does this calculator test?

It studies positive-term series by comparing them to a benchmark series. It reports whether the chosen comparison supports convergence, divergence, or remains inconclusive.

2) When should I use direct comparison?

Use direct comparison when you can show one series is eventually smaller or larger than a familiar benchmark. It is strongest when the inequality is obvious from the denominator powers.

3) When is limit comparison better?

Use limit comparison when both series have similar dominant growth or decay. A finite positive ratio is the cleanest case because it immediately gives the same convergence behavior.

4) Why must the terms be positive?

Standard comparison tests rely on positive terms for large n. Sign-changing series need different tests such as absolute convergence, alternating series checks, or more specialized methods.

5) What benchmark should I choose?

Choose a benchmark whose behavior you already know, such as 1/n^p, 1/[n(ln n)^q], or a geometric-decay series. The best benchmark matches the dominant denominator structure.

6) Does a ratio limit of zero always prove convergence?

No. A ratio limit of zero proves convergence only when the benchmark series is convergent. If the benchmark diverges, the conclusion may still be inconclusive.

7) Can I use decimal exponents?

Yes. The calculator accepts decimal values for p and q, which is useful for asymptotic models and generalized power comparisons in advanced coursework.

8) Why does g > 1 make the series converge?

A denominator factor g^n with g greater than 1 creates exponential decay. Exponential decay dominates polynomial and logarithmic factors, forcing the terms to shrink rapidly.