Conditionally convergent series

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

Conditional convergence is fragile under rearrangements : the Riemann rearrangement theorem shows that rearranging terms can change the sum or destroy convergence. Many standard examples are produced using the alternating series test .

Examples:

• The alternating harmonic series $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}$ converges conditionally.
• The alternating series $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{\sqrt{n}}$ converges conditionally.