Convergence or Divergence of a Series Calculator

Calculator Inputs

Series term a(n)

Use n as the index. Supported functions include sqrt, log, ln, exp, sin, cos, tan, abs, pow, min, and max.

Test method

Start index n

Terms to sample

Tolerance

p value for direct p-series

Comparison p value

Geometric first term

Geometric common ratio

Rows to show

Example Data Table

Series Term	Suggested Test	Expected Result	Main Reason
1/n^2	p-series	Convergent	p is greater than 1
1/n	p-series	Divergent	p equals 1
(1/2)^n	Geometric	Convergent	\|r\| is less than 1
2^n/n	Ratio test	Divergent	Ratio is greater than 1
(-1)^(n+1)/n	Alternating	Conditionally convergent	Terms alternate and decrease

Formula Used

nth-term test: if lim a(n) is not zero, then the series diverges.

Ratio test: L = lim |a(n+1) / a(n)|. The series converges when L < 1 and diverges when L > 1.

Root test: L = lim nth root of |a(n)|. The series converges when L < 1 and diverges when L > 1.

p-series: sum 1 / n^p converges when p > 1 and diverges when p <= 1.

Geometric series: sum a r^n converges when |r| < 1. Its sum is a / (1 - r), using the selected first term.

Alternating test: an alternating series converges when term magnitudes decrease and approach zero.

How to Use This Calculator

Enter a formula for a(n), using n as the running index. Select Auto review for a broad check, or choose a specific test when your course requires one. Set the starting index, sample size, tolerance, and table length. Press Calculate. The decision appears above the form. Use CSV or PDF to save the result.

Understanding Series Convergence

An infinite series adds terms without a final stopping point. The main question is simple. Does the running total settle near one finite number, or does it move away forever? This calculator helps by testing the sequence term a(n), building partial sums, and comparing tail behavior.

Why the First Check Matters

The nth term test is the first filter. A series cannot converge when its terms fail to approach zero. This rule does not prove convergence by itself. It only proves divergence when the term limit is not zero. That is why the calculator also reviews ratio, root, comparison, and alternating behavior.

Ratio and Root Tests

The ratio test is useful for factorials, powers, and exponential patterns. It studies the size of one term compared with the next term. A limiting ratio below one suggests absolute convergence. A value above one suggests divergence. The root test is similar, but it works well when every term contains a power of n.

Special Series Forms

Some series are easier to judge directly. A p-series depends on the exponent p. It converges only when p is greater than one. A geometric series depends on the common ratio. It converges when the absolute ratio is less than one. Alternating series need a separate view. Their signs switch, their magnitudes decrease, and the terms approach zero.

Numerical Limits

No numerical checker replaces a formal proof. Large samples can still miss unusual behavior. A function may change after the tested range. Rounding can also affect very small terms. Use the result as a strong guide, then write the final proof using the named test. Increase the sample size when the answer is close or unclear.

Practical Study Value

The table shows terms and partial sums together. This makes patterns easier to see. A stable partial sum suggests convergence. Fast growth suggests divergence. Exporting the table helps with homework notes, worksheets, and class demonstrations. The best workflow is to test, inspect, then explain each step clearly in your own words with confidence.

FAQs

1. What does convergence mean?

Convergence means the infinite sum approaches one finite value as more terms are added. The partial sums get closer to a stable limit.

2. What does divergence mean?

Divergence means the infinite sum does not settle at a finite value. It may grow, oscillate, or fail a required convergence condition.

3. Which test should I choose?

Use Auto review for a quick study check. Choose ratio for factorials or exponentials, root for powers, p-series for 1/n^p, and alternating for sign-changing terms.

4. Can this calculator prove every series?

No. It gives a numerical and rule-based guide. Some series require deeper symbolic work, special comparisons, or a formal proof from a textbook method.

5. Why is the result sometimes inconclusive?

Some sampled limits are too close to one, or the partial sums change slowly. More terms may help, but some cases still need manual proof.

6. What expression format is allowed?

Use n as the variable. You may enter operations like +, -, *, /, ^, parentheses, and functions such as sqrt, log, exp, sin, cos, and abs.

7. What is absolute convergence?

Absolute convergence means the series formed from absolute values also converges. It is stronger than conditional convergence and often follows from ratio or root tests.

8. Why export the result?

CSV is useful for spreadsheets and further checking. PDF is useful for saving a clean report with the decision, formulas, and visible term rows.