Give an example of a series that is conditionally convergent (that is, convergent but not absolutely convergent). Infinite Series, Convergence tests,
\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...
is convergent by the alternating series test (the terms are alternately positive and negative, and their magnitudes decrease to zero). But
\sum_{n=1}^{\infty}| \frac{(-1)^{n+1}}{n} | =\sum_{n=1}^{\infty}\frac{1}{n}
diverges.
