Series Convergence Test Calculator

Test whether an infinite series converges using the Ratio, Root, Integral, Comparison, Limit Comparison, Alternating Series and p-Series tests, with step-by-step solutions and animated partial sum graphs.

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About Series Convergence Test Calculator

The Series Convergence Test Calculator is a comprehensive tool for determining whether an infinite series converges or diverges. It systematically applies multiple convergence tests — including the Ratio Test, Root Test, Integral Test, Alternating Series Test, Comparison Tests, and more — to provide a definitive answer with step-by-step mathematical reasoning.

Available Convergence Tests

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Ratio Test

Examines lim |aₙ₊₁/aₙ| to determine convergence

√

Root Test

Examines lim |aₙ|^(1/n) for convergence

∫

Integral Test

Compares series to an improper integral

Alternating Series

Leibniz criterion for alternating series

⇔

Comparison Tests

Direct and Limit Comparison with known series

≠0

Divergence Test

If lim aₙ ≠ 0, the series must diverge

Understanding Series Convergence

An infinite series \(\sum_{n=1}^{\infty} a_n\) converges if the sequence of partial sums \(S_N = \sum_{n=1}^{N} a_n\) approaches a finite limit as \(N \to \infty\). If no such limit exists, the series diverges. Determining convergence is a fundamental problem in calculus and analysis, and several tests have been developed to handle different types of series.

Convergence Test Decision Flowchart

Test	When to Use	Conclusion
Divergence Test	Always check first	If \(\lim a_n \neq 0\), series diverges
Geometric Series	Series of the form \(\sum r^n\)	Converges iff \(\|r\| < 1\)
p-Series Test	Series of the form \(\sum 1/n^p\)	Converges iff \(p > 1\)
Ratio Test	Series with factorials, exponentials	\(L < 1\): converges; \(L > 1\): diverges
Root Test	Series with nth powers	\(L < 1\): converges; \(L > 1\): diverges
Integral Test	Positive, decreasing terms	Series and integral converge/diverge together
Alternating Series Test	Alternating sign series	Converges if \(\|a_n\|\) decreasing → 0
Limit Comparison	Compare with known series	Both converge or both diverge if \(0 < L < \infty\)

Absolute vs. Conditional Convergence

A series \(\sum a_n\) converges absolutely if \(\sum |a_n|\) also converges. It converges conditionally if \(\sum a_n\) converges but \(\sum |a_n|\) diverges. Absolute convergence is stronger — any absolutely convergent series is also convergent, but not vice versa. The classic example of conditional convergence is the alternating harmonic series \(\sum (-1)^{n+1}/n\).

How to Use the Series Convergence Test Calculator

Select a series type from the dropdown menu (p-Series, Geometric, Alternating, etc.) or click a quick example button.
Enter the required parameters for your chosen series. For example, enter p = 2 for the series \(\sum 1/n^2\).
Set the number of terms (5–100) for the partial sum visualization. More terms give a clearer picture of convergence behavior.
Click "Test Convergence" to run all applicable tests simultaneously.
Review the results: the verdict banner, individual test breakdowns (click to expand), the first terms table, and the interactive partial sum graph.

Frequently Asked Questions

What is the Ratio Test for series convergence?

The Ratio Test examines the limit \(L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\). If \(L < 1\), the series converges absolutely. If \(L > 1\) or \(L = \infty\), the series diverges. If \(L = 1\), the test is inconclusive. The Ratio Test is especially effective for series involving factorials and exponential terms.

When should I use the Root Test vs the Ratio Test?

The Root Test is often more effective for series involving nth powers, like \(a^n\) or \(n^n\). The Ratio Test works better for series with factorials, like \(n!\) or products of consecutive integers. Both tests are equivalent in many cases, but one may be significantly easier to compute than the other for a given series.

What is conditional convergence?

A series converges conditionally if it converges but does not converge absolutely. This means the sum of the original series is finite, but the sum of the absolute values of the terms is infinite. The alternating harmonic series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) is the classic example — it converges to \(\ln 2\), but the harmonic series \(\sum 1/n\) diverges.

How does the Integral Test work?

The Integral Test states that if \(f(x)\) is positive, continuous, and decreasing for \(x \geq 1\), and \(a_n = f(n)\), then \(\sum a_n\) and \(\int_1^{\infty} f(x)\,dx\) either both converge or both diverge. It is particularly useful for p-series and logarithmic series where other tests may be inconclusive.

What is the Alternating Series Test?

The Alternating Series Test (also called the Leibniz Test) applies to series of the form \(\sum (-1)^n b_n\) where \(b_n > 0\). If \(b_n\) is decreasing and \(\lim_{n \to \infty} b_n = 0\), the series converges. Note that this test only establishes convergence, not absolute convergence — the series may still be conditionally convergent.

Reference this content, page, or tool as:

"Series Convergence Test Calculator" at https://MiniWebtool.com/series-convergence-test-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-06

Developer API available: Run this tool from your app, automation, or agent with one JSON HTTP request. View API docs

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