How does the alternating-series test estimate the truncation error?

Conditions

• The series satisfies the conditions of the alternating-series test.
• The error is estimated after summing $n$ terms.

Reasoning, step by step

1. Sum the first $n$ terms of the alternating series.
2. Identify the $(n+1)$-th term of the series.
3. Calculate the absolute value of the $(n+1)$-th term.
4. Use this value as an upper bound for the truncation error $|R_n|$.

Example

For the series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$, if you stop after $n$ terms, the error $|R_n|$ is at most $\frac{1}{n+1}$.

Common misconceptions

• Assuming the error bound applies to non-alternating series.
• Believing the error is exactly equal to the next term, rather than bounded by it.

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Connected concepts

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• Why is the error bounded by the next term?
• Does this error bound hold if the terms are not monotonically decreasing?