Test series quickly with limit ratios, sample terms, and clear outcomes. Compare common patterns easily. See convergence steps, edge cases, and printable summaries instantly.
The ratio test studies the limit
L = lim n→∞ |aₙ₊₁ / aₙ|
For the model aₙ = c · n^p · b^n, the limit becomes |b|.
If (n!)^q is in the denominator, the limit becomes 0.
If (n!)^q is in the numerator, the limit grows without bound.
| Series Term aₙ | Ratio Limit L | Outcome |
|---|---|---|
| 1 / n! | 0 | Converges absolutely |
| 2^n / n! | 0 | Converges absolutely |
| n! / 3^n | ∞ | Diverges |
| 1 / n^2 | 1 | Inconclusive by ratio test |
| (4/5)^n | 4/5 | Converges absolutely |
The ratio test is a fast way to study infinite series. It works well for terms with powers, exponentials, and factorials. Many textbook problems fit this pattern. The test compares consecutive terms. That comparison shows how quickly the series terms shrink or grow.
This calculator helps when a series looks hard at first glance. It reduces the work to a clean ratio. It is useful for expressions such as n^p, b^n, n!, and combinations of them. These forms appear often in calculus, mathematical analysis, probability, and applied models.
The core idea is simple. Find the limit of |aₙ₊₁ / aₙ|. If that limit is less than one, the terms decay fast enough, and the series converges absolutely. If the limit is greater than one, the terms do not decay enough, and the series diverges. If the limit equals one, the ratio test cannot decide. You then switch to another method.
This page supports direct term input and a structured model. The model is helpful for common families. A plain exponential term usually gives a limit equal to the base magnitude. A factorial in the denominator drives the limit toward zero. A factorial in the numerator drives the ratio upward and usually forces divergence.
The worked output gives a sample ratio, the limiting value, and a clear decision line. That makes it useful for homework review, exam practice, and concept checks. The example table also shows where the test becomes inconclusive. This matters because many learners expect one method to solve every series. It does not. A good calculator should show both answers and limits.
The best workflow is simple. Identify the term pattern. Enter it. Read the limit. Then read the conclusion. If the result says inconclusive, move to another convergence test. That is still progress. It tells you exactly what to try next.
It checks the limit of the absolute ratio of consecutive terms. That limit shows whether the series terms shrink fast enough for absolute convergence.
It is most useful for series with factorials, exponentials, powers, or products of these forms. Those patterns simplify well under consecutive term division.
If the ratio limit is below 1, the series converges absolutely. The terms decay rapidly enough to make the infinite sum stable.
If the ratio limit is above 1, the series diverges. The terms do not decrease enough, or they grow, so the series cannot settle.
The ratio test is inconclusive. You need another method, such as comparison, root, integral, alternating series, or p-series testing.
A denominator factorial grows very quickly. That growth makes consecutive ratios smaller, often pushing the limit to zero and forcing convergence.
Not always. Direct mode shows a sample ratio. The formal ratio test still needs the limit as n approaches infinity unless you already know it.
Exports help with notes, assignments, and revision sheets. You can save the decision, sample values, and example table for later review.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.