Alternating Series Tester

Math Calculus • Infinite Series and Sequences

Written by STEM Calculators Team Published December 30, 2025 Updated February 24, 2026

7. Alternating Series Tester

Tests the Leibniz alternating series test and computes the remainder bound \( |R_N| \le a_{N+1} \) when the hypotheses hold. Includes a zoomable, pannable partial-sum plot and a “how many terms for \( \varepsilon \)” helper.

Inputs

Input mode

Recommended: Leibniz form \(t_n = (-1)^{n+s}a_n\) with \(a_n \ge 0\).

Term input

Variable: n. Constants: pi, e. Functions: sin cos tan asin acos atan sqrt abs exp ln/log. Use ^ for powers.

Sign pattern

In Leibniz mode, this sets the alternating sign of \(t_n\). In general mode, “Auto” tries to pick \(s\in\{0,1\}\).

Start index \(n_0\)

Approximation terms \(N\)

Computes \(S_N=\sum_{n=n_0}^{n_0+N-1} t_n\) and (if applicable) \(|R_N|\le a_{N+1}\).

Plot terms

Number of partial sums shown on the graph (≤ 500).

Target \( \varepsilon \)

Find smallest \(N\le N_{\max}\) with \(a_{N+1}\le\varepsilon\).

Search cap \(N_{\max}\)

Options

Show steps Sample terms table Absolute test (heuristic) Auto-solve on change

Presets

Click a preset to load & compute.

Ready

Graph

Drag to pan • wheel/pinch to zoom. Plots partial sums \(S_k\) versus index \(k\).

Graph toggles

Plot partial sums Shade \([S_k,S_{k+1}]\) Legend Auto-fit after solve

x: 0, y: 0, zoom(px/unit): 60

Results & steps

Enter a series definition and click Calculate.

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Theory: Alternating Series Tester

1) Alternating series (Leibniz form)

A common alternating series form is \[ \sum_{n=n_0}^{\infty} (-1)^{n+s} a_n, \] where \(a_n \ge 0\) and \(s\in\{0,1\}\) selects whether the first term is positive or negative.

The partial sums \(S_N\) typically oscillate around the true sum because the signs alternate.

2) Leibniz alternating series test

If the sequence \(\{a_n\}\) satisfies \[ a_n \ge 0,\qquad a_{n+1}\le a_n\ \text{(eventually)},\qquad \lim_{n\to\infty} a_n = 0, \] then the alternating series \(\sum (-1)^{n+s}a_n\) converges.

The condition \(a_n\to 0\) is necessary: if \(a_n\not\to 0\), then the series diverges (term test).

3) Error bound (remainder estimate)

Let \(S\) be the exact sum and \[ S_N=\sum_{n=n_0}^{n_0+N-1} (-1)^{n+s}a_n \] the partial sum using \(N\) terms. Under the Leibniz hypotheses, \[ |S-S_N| \le a_{N+1}. \]

This is very useful: to guarantee error below \(\varepsilon\), it is enough to choose \(N\) so that \[ a_{N+1} \le \varepsilon. \]

Moreover, the true sum lies between consecutive partial sums: \[ S \in [S_N,\, S_{N+1}]. \]

4) Conditional vs absolute convergence

A series \(\sum t_n\) is absolutely convergent if \(\sum |t_n|\) converges. It is conditionally convergent if \(\sum t_n\) converges but \(\sum |t_n|\) diverges.

Example: \[ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=\ln 2 \] converges by Leibniz, but \(\sum_{n=1}^{\infty}\frac{1}{n}\) diverges, so it is conditional.

5) University note

Some alternating-looking series are not monotone in \(a_n\). More advanced tools include Dirichlet’s test and Abel’s test.

Frequently Asked Questions

What is the Leibniz alternating series test?

For a series sum (-1)^(n+s) a_n with a_n >= 0, if a_n is eventually decreasing and a_n -> 0, then the series converges. If a_n does not approach 0, the series diverges by the n-th term test.

How is the alternating-series remainder bound computed?

When the Leibniz hypotheses hold, the truncation error satisfies |S - S_N| <= a_(N+1), where S_N is the partial sum using N terms starting at n0. The true sum lies between consecutive partial sums.

How many terms do I need to get error below epsilon?

Under the Leibniz conditions, it is enough to choose N so that a_(N+1) <= epsilon. The calculator can search up to N_max and report the smallest N that meets this threshold.

Why can an alternating series be inconclusive in this tester?

If the tool cannot confirm eventual decrease of a_n or cannot verify that a_n -> 0 over the tested range, the Leibniz test may not apply. Some alternating-looking series require other tests such as Dirichlet or Abel.

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Frequently Asked Questions

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