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Conditionally convergent series

A convergent series that is not absolutely convergent.

A conditionally convergent series is a series ∑n=1∞an\sum_{n=1}^\infty a_n that is but not , meaning that ∑n=1∞an\sum_{n=1}^\infty a_n converges while ∑n=1∞∣an∣\sum_{n=1}^\infty |a_n| diverges.

Examples
  • The alternating harmonic series ∑n=1∞(−1)n−1n\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n} converges conditionally.
  • The alternating series ∑n=1∞(−1)n−1n\sum_{n=1}^\infty \frac{(-1)^{n-1}}{\sqrt{n}} converges conditionally.
Remarks

Conditional convergence is fragile under : the shows that rearranging terms can change the sum or destroy convergence. Many standard examples are produced using the .