What This Calculator Does
A limit comparison test checks two positive term series. It compares the target term a_n with a known benchmark term b_n. The main value is the limit of a_n divided by b_n as n grows. When that limit is positive and finite, both series share the same convergence behavior.
Why It Helps
Many series look difficult at first. Their terms may contain powers, roots, logarithms, exponentials, or mixed factors. A direct test can be slow. Limit comparison reduces the problem to a simpler benchmark. Common benchmarks include p-series, geometric series, harmonic series, and logarithmic series.
Advanced Input Control
This calculator lets you enter custom formulas for both terms. You can choose absolute ratio mode for positive term analysis. You can also set the known behavior of the comparison series. The tool samples larger n values and estimates the limit numerically. It then gives a theorem-based conclusion when the evidence is strong enough.
Interpreting Results
A finite positive limit means the two series rise or fall together. If the benchmark converges, the target converges. If the benchmark diverges, the target diverges. A zero limit can prove convergence only when the benchmark converges. An infinite limit can prove divergence only when the benchmark diverges. Other cases need another test.
Best Practices
Always choose b_n from the dominant part of a_n. Match the largest powers, main logarithms, or strongest exponentials. Start at an index where terms are defined and positive. For logarithmic expressions, n often starts at 2 or 3. Increase precision when ratios change slowly.
Numerical Caution
This page uses numerical sampling, not symbolic algebra. It helps guide decisions, but it cannot replace a formal proof. Some limits converge very slowly. Oscillating or sign-changing terms can also mislead the ratio table. Use the displayed ratio trend, stability score, and theorem notes together.
Practical Use
The calculator is useful for homework checking, teaching notes, and quick series screening. It also exports ratio data. The CSV file supports spreadsheet review. The PDF report helps save a clean summary. You can compare several benchmarks and choose the clearest proof path. Store each trial with the same settings. This keeps your convergence argument organized and easier to explain during revision or grading.