Infinite Series Ratio Test Calculator

Calculator Inputs

Series Label

Quick Preset Choose a preset 1 / n! (2^n) / n! n! / (2^n) 1 / n^2 (3^n) / (5^n) n / (3^n) (n^2) / (2^n) n! / n^n

Term Expression a_n

Start n

Sample Count

Tail Values for Limit Estimate

Decision Tolerance

Display Decimals

Optional Notes

Expression tips: use n, ^, !, sqrt(), exp(), abs(), pi, and e.

Examples: 1/(n!), (2^n)/(n!), n!/(2^n), 1/(n^2).

Compute Ratio Test Reset

Example Data Table

Formula Used

The ratio test studies the limit below:

L = lim n→∞ |a(n+1) / a(n)|

• If L < 1, the series converges absolutely.
• If L > 1 or the limit grows without bound, the series diverges.
• If L = 1, the test is inconclusive.

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This calculator estimates the limit numerically by sampling ratios for large values of n, then averaging the most recent valid ratios.

How to Use This Calculator

1. Enter a formula for the general term a_n.
2. Choose a starting value for n.
3. Set how many sample ratios you want to inspect.
4. Choose how many tail ratios should estimate the limit.
5. Set the tolerance for classifying the result.
6. Press Compute Ratio Test.
7. Review the summary, graph, and computed sample table.
8. Export the results as CSV or PDF if needed.

About the Ratio Test

The ratio test is useful for many exponential, factorial, and power-based series. It examines how consecutive terms compare as the index grows. When that ratio settles below one, terms shrink rapidly enough for convergence. When it settles above one, the series cannot converge.

This page uses sampled numerical values instead of symbolic proof. That makes it practical for checking custom formulas fast, especially when you want a quick diagnostic before formal work. The limit estimate is based on recent ratio values, so larger sample sizes often improve confidence.

Factorials, exponentials, and mixed expressions usually respond well to ratio testing. Some algebraic series, including many p-series, produce a ratio limit of one. In those cases, the ratio test does not fail mathematically. It simply says another convergence test is required.

The computed table shows each sampled index, the term value, its magnitude, the next term, and the absolute ratio. The first graph tracks ratio behavior across sampled indices. The second graph tracks absolute term size, helping you see whether terms decay toward zero or remain too large.

Use the CSV export when you want to inspect values in a spreadsheet. Use the PDF export when you need a report snapshot for notes, teaching, or review. The calculator is intended for learning, checking, and pattern recognition. For formal proofs, always confirm borderline cases with an exact analytical argument.

FAQs

1) What does this calculator test?

It estimates the ratio-test limit for an infinite series with general term a_n. Then it classifies the series as convergent, divergent, or inconclusive from that estimated limit.

2) When is the ratio test most useful?

It works especially well for series involving factorials, exponentials, powers, and combinations of those patterns. These often produce a clean ratio limit away from one.

3) What happens if the estimated limit equals 1?

The result is inconclusive. That means the ratio test alone cannot decide convergence. You should try another method, such as comparison, root, integral, or alternating-series testing.

4) Why is the result numerical instead of symbolic?

This page evaluates sampled terms and ratios directly in the browser. That makes it flexible for custom inputs, but the output should be treated as a strong estimate rather than formal proof.

5) Can I enter factorial expressions?

Yes. Use the exclamation mark, such as 1/(n!) or (2^n)/(n!). The expression engine also supports powers, roots, exponential functions, and common constants.

6) Why do some values become very large?

Factorials and exponentials can grow or shrink extremely fast. Large or tiny values may appear in scientific notation so the table and summary stay readable.

7) What does tolerance change?

Tolerance controls how close the estimated limit may be to one before the calculator marks the result inconclusive. Smaller tolerance gives stricter classification near the boundary.

8) Are CSV and PDF exports included?

Yes. After calculation, you can export the sampled table and summary as CSV or PDF. That is helpful for assignments, checking work, or keeping reference notes.