What are the sufficient conditions for the alternating-series test to guarantee convergence?

Conditions

• The series is an alternating series.
• The absolute values of the terms are nonnegative.
• The sequence of absolute values is monotonically decreasing.
• The limit of the terms as n approaches infinity is zero.

Reasoning, step by step

1. Identify the general term of the alternating series.
2. Check if the absolute value of the general term is monotonically decreasing.
3. Verify if the limit of the general term as n approaches infinity is zero.
4. Conclude that if both conditions are satisfied, the series converges by the alternating-series test.

Example

For the series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$, the absolute values $\frac{1}{n}$ decrease monotonically to zero, satisfying the conditions for convergence.

Common misconceptions

• Believing that monotonicity is necessary for every convergent alternating series.
• Thinking that a finite plot of partial sums alone proves convergence.

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Connected concepts

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• Is monotonicity necessary for the convergence of all alternating series?
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