Direct Comparison Test Calculator

Calculator Form

Target term a_n

Use n, +, -, *, /, ^, sqrt, log, exp, abs.

Comparison term b_n

Example: 1/(n^2), 1/n, or 0.5^n.

Known result for b_n

Comparison direction

Multiplier C

Starting n

Ending n

Tolerance

Tail ratio window

Formula Used

The direct comparison test uses positive terms after some index N.

If 0 ≤ a_n ≤ Cb_n and Σb_n converges, then Σa_n converges.

If a_n ≥ Cb_n ≥ 0 and Σb_n diverges, then Σa_n diverges.

The calculator also reports Σa_n and Σb_n over the selected finite range.

How to Use This Calculator

Enter the target term as a formula in n. Enter a simpler comparison term. Choose whether that comparison series is convergent, divergent, or unknown. Select an automatic direction, or force an upper or lower bound. Add a positive multiplier C if the bound needs scaling. Then calculate and review the result above the form.

Example Data Table

Target a_n	Comparison b_n	Known result	Useful inequality	Conclusion
1/(n²+n)	1/n²	Convergent	a_n ≤ b_n	Convergent
1/√n	1/n	Divergent	a_n ≥ b_n	Divergent
1/(n²+5)	1/n²	Convergent	a_n ≤ b_n	Convergent

Understanding Direct Comparison

The direct comparison test is a practical tool for positive series. It compares a difficult series with a known series. The known series becomes a benchmark. If the harder terms stay below a convergent benchmark, the harder series also converges. If they stay above a divergent benchmark, it also diverges.

Why Positivity Matters

The test needs nonnegative terms after some starting index. This condition keeps partial sums ordered. Without that order, an upper or lower bound may not control the total behavior. Many useful examples satisfy positivity. These include rational terms, p series, geometric tails, and many probability style sequences.

Choosing a Good Benchmark

A good comparison term should be simpler than the target term. It should also have a known result. The p series 1 divided by n to the p power is common. It converges when p is greater than one. It diverges when p is one or less. Geometric series are also useful. They converge when the absolute ratio is below one.

Reading the Calculator Results

This calculator samples terms across a selected index range. It checks positivity, ratios, bounds, partial sums, and direct comparison logic. The result should guide reasoning, not replace proof. A sampled inequality is strong evidence. A formal solution still needs an algebraic bound that holds for all large n.

Practical Study Tips

Start by simplifying the leading behavior of each term. Ignore lower order parts when choosing the benchmark. Then check the exact inequality. Use a constant multiplier when one series is only smaller after scaling. This often makes comparison possible.

Common Mistakes

Do not conclude convergence from being below a divergent series. Do not conclude divergence from being above a convergent series. Those cases are inconclusive. Also avoid using the test when terms change sign. Use another method for alternating or signed series.

Proof Workflow

Write the target term first. State the benchmark next. Prove the inequality with clear algebra. Mention the benchmark result. Then state the conclusion with the comparison direction.

Final Note

Direct comparison is clear because it uses order. It turns one hard question into a known one. With careful bounds, it becomes one of the most reliable convergence tests for routine series work today.

FAQs

What is the direct comparison test?

It is a convergence test for nonnegative series. It compares a target series with a known series. The conclusion follows when the correct upper or lower inequality holds eventually.

Can this calculator prove convergence fully?

It gives strong computational guidance. A formal proof still needs algebra showing the inequality holds for every sufficiently large n.

What does the multiplier C mean?

C is a positive scaling constant. It lets you test bounds like aₙ ≤ Cbₙ or aₙ ≥ Cbₙ when the terms differ by a fixed factor.

When is the result convergent?

If 0 ≤ aₙ ≤ Cbₙ and the comparison series converges, the target series converges by direct comparison.

When is the result divergent?

If aₙ ≥ Cbₙ ≥ 0 and the comparison series diverges, the target series diverges by direct comparison.

Why can a result be inconclusive?

The inequality may point the wrong way. Being smaller than a divergent series proves nothing. Being larger than a convergent series also proves nothing.

Which formulas can I enter?

You can use n, arithmetic operators, powers, parentheses, and common functions like sqrt, log, exp, abs, sin, and cos.

Does the test work with negative terms?

No. Direct comparison needs nonnegative terms after some index. For alternating or signed series, use a different convergence test.