Convergence Tester

Math Algebra • Sequences and Series

Written by STEM Calculators Team Published December 19, 2025 Updated February 24, 2026

6. Convergence Tester

Determine convergence of a series \(\sum a_n\) using ratio, root, p-series recognition, (limit) comparison, and alternating series checks. Includes partial-sum graph.

Term \(a_n\) (use variable n): Start index \(n_0\): Test selector:

Detect “alternating” factor automatically (e.g., \((-1)^n\)) When alternating, also test absolute convergence Show term/partial-sum table Show point labels on graph Auto label density

Partial sums up to \(N\) (max 2000):

Comparison series \(b_n\): Parameter \(p\) (if \(1/n^p\)): Parameter \(r\) (if \(r^n\)):

Ready

Partial sums \(S_N=\sum_{n=n_0}^{N} a_n\)

x-axis: N • y-axis: S_N. Drag to pan • wheel/pinch to zoom.

N: 0, S: 0 sx: 40, sy: 40 Partial sums \(S_N\)

Click Test convergence to see results and steps.

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Theory: Series Convergence Tests

A series \(\sum_{n=n_0}^{\infty} a_n\) converges if its partial sums \(S_N=\sum_{n=n_0}^{N} a_n\) approach a finite limit as \(N\to\infty\). If the partial sums grow without bound or oscillate without settling, the series diverges.

1) Absolute vs conditional convergence

Absolute convergence: \(\sum |a_n|\) converges \(\Rightarrow \sum a_n\) converges.
Conditional convergence: \(\sum a_n\) converges but \(\sum |a_n|\) diverges (common for alternating series).

2) Ratio test

\[ L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|. \]

If \(L<1\), then \(\sum a_n\) converges absolutely.
If \(L>1\) (or \(L=\infty\)), the series diverges.
If \(L=1\), the test is inconclusive (you need another test).

3) Root test

\[ L=\lim_{n\to\infty}\sqrt[n]{|a_n|}. \]

The conclusions are the same as the ratio test: \(L<1\) gives absolute convergence, \(L>1\) gives divergence, \(L=1\) is inconclusive.

4) p-series and geometric series (fast recognition)

p-series: \(\displaystyle \sum \frac{1}{n^p}\)

Converges if \(p>1\)
Diverges if \(p\le 1\)

Geometric series: \(\displaystyle \sum ar^{n}\)

Converges if \(|r|<1\), with sum \(\frac{a r^{n_0}}{1-r}\) when applicable
Diverges if \(|r|\ge 1\)

5) Limit comparison test

Pick a known comparison series \(\sum b_n\) with \(b_n>0\). Compute

\[ L=\lim_{n\to\infty}\frac{a_n}{b_n}\quad (\text{often use }|a_n| \text{ for absolute comparison}). \]

If \(0same convergence behavior.

6) Alternating series test (Leibniz)

For an alternating series \(\sum (-1)^n b_n\) (or \((-1)^{n+1}b_n\)) with \(b_n\ge 0\):

If \(b_n\) is eventually decreasing, and
\(\lim_{n\to\infty} b_n = 0\),

then the series converges (often conditionally).

7) Raabe’s test (advanced)

When ratio/root tests give \(L=1\), Raabe’s test can help for positive terms:

\[ R=\lim_{n\to\infty} n\left(1-\frac{a_{n+1}}{a_n}\right). \]

If \(R>1\), the series converges.
If \(R<1\), the series diverges.
If \(R=1\), inconclusive.

Worked example: \(\sum \frac{1}{n^2}\)

This is a p-series with \(p=2>1\), so it converges. Ratio and root tests both typically produce limits equal to \(1\) here, so they are inconclusive—this is normal.

Frequently Asked Questions

What does it mean for a series to converge?

A series converges when its partial sums S_N approach a finite limit as N goes to infinity. If the partial sums grow without bound or fail to settle, the series diverges.

How does the ratio test decide convergence?

It computes L = lim n->infinity |a_{n+1}/a_n|. If L < 1 the series converges absolutely, if L > 1 it diverges, and if L = 1 the result is inconclusive.

When is a p-series convergent?

A p-series has the form sum 1/n^p. It converges when p > 1 and diverges when p <= 1.

Why can an alternating series converge even if the absolute series diverges?

An alternating series can converge conditionally when terms decrease in magnitude and approach 0, even if sum |a_n| diverges. The calculator can check alternating behavior and also test absolute convergence when enabled.

What does the partial-sum graph show?

It plots S_N = sum_{n=n_0}^N a_n versus N. Convergence typically appears as S_N leveling off toward a stable value, while divergence often shows unbounded growth or persistent oscillation.

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