Series Comparison Test Calculator

Calculator Inputs

Tested term a_n

Example: 1/(n*n + 2*n)

Benchmark term b_n

Example: 1/(n*n)

Benchmark name

Known benchmark behavior

Test mode

Starting n

Ending n

Sample count

Tolerance

Example Data Table

Example	a_n	b_n	Known benchmark	Expected decision
Rational comparison	1/(nn + 2n)	1/(n*n)	Convergent p-series	Convergent by comparison
Harmonic style	(n+1)/(n*n)	1/n	Divergent harmonic series	Divergent by limit comparison
Geometric comparison	1/pow(3,n)	1/pow(2,n)	Convergent geometric series	Convergent by direct comparison

Formula Used

Direct comparison for convergence: If 0 ≤ a_n ≤ b_n and ∑b_n converges, then ∑a_n converges.

Direct comparison for divergence: If 0 ≤ b_n ≤ a_n and ∑b_n diverges, then ∑a_n diverges.

Limit comparison: L = lim_n→∞ a_n / b_n. If 0 < L < ∞, both positive series share the same convergence behavior.

The calculator estimates L using later sampled ratios. It also checks sampled inequalities for direct comparison.

How To Use This Calculator

Enter the tested term using n as the variable.
Enter a benchmark term with known behavior.
Select whether the benchmark converges, diverges, or is unknown.
Choose direct comparison, limit comparison, or both.
Set the n range and sample count for numerical review.
Press Calculate to show results below the header.
Use CSV or PDF buttons to save the report.

Understanding The Series Comparison Test

A series comparison test helps decide whether an infinite series converges or diverges. It works best with positive terms. The idea is simple. A difficult series is compared with a familiar benchmark series. Common benchmarks include p series and geometric series. If the benchmark behavior is known, the new series may inherit a decision.

Why Comparison Matters

Many series cannot be summed directly. Their partial sums may be hard to simplify. Comparison gives a practical path. It checks term size instead of exact totals. Smaller positive terms can be controlled by a convergent benchmark. Larger positive terms can confirm divergence when the benchmark already diverges. This makes the method useful in calculus, analysis, and applied modeling.

Direct Comparison Method

Direct comparison uses inequalities. For convergence, show that zero is less than or equal to a term, and that term is less than or equal to a convergent benchmark term. Then the tested series converges. For divergence, show that the tested term is greater than or equal to a divergent benchmark term. Then the tested series diverges. The calculator samples values and reports observed inequalities. Formal proof still needs algebra.

Limit Comparison Method

Limit comparison studies the ratio between two positive terms. If the ratio approaches a positive finite number, both series share the same behavior. If the ratio approaches zero, the tested series may converge when the benchmark converges. If the ratio grows without bound, the tested series may diverge when the benchmark diverges. This tool estimates that ratio using later sample points.

Good Input Choices

Choose a benchmark that matches the leading behavior of the tested term. For rational expressions, compare the highest powers of n. For factorials or exponentials, choose a term with similar growth. For logarithmic terms, include the logarithm in the benchmark when needed. Use a larger ending value when early terms distort the ratio.

Interpreting Results

The answer is a guided calculation, not a complete proof. Numeric sampling can miss unusual behavior. Always check positivity and long term inequality. Use the export tools to save the data. Review ratios, bounds, and decisions before using the conclusion in homework, reports, or engineering notes. Compare several benchmarks when the first match fails.

FAQs

1. What is the series comparison test?

It is a convergence method for positive term series. It compares a difficult series with a simpler benchmark whose behavior is already known.

2. Which benchmark should I choose?

Choose a benchmark that matches the dominant part of the term. Power, geometric, logarithmic, and factorial patterns are common choices.

3. Does this calculator prove convergence?

It gives a numerical guide and structured evidence. A formal proof still needs algebraic inequalities or a symbolic limit.

4. Why must terms be positive?

Standard comparison tests depend on ordered positive terms. Negative or sign-changing terms may need absolute convergence tests or alternating series methods.

5. What does the ratio estimate mean?

It approximates a_n divided by b_n for large n. A positive finite estimate supports the limit comparison test.

6. Can I use trigonometric functions?

Yes. Supported functions include sin, cos, tan, log, ln, log10, exp, sqrt, abs, pow, min, and max.

7. Why is my result inconclusive?

The sampled terms may not show the needed inequality or ratio behavior. Try a better benchmark, wider range, or symbolic work.

8. What exports are available?

You can download a CSV table or a simple PDF report. Both include inputs, ratios, decisions, and sampled values.