Difficulties to understand series Hi in my book there is a series:
$$\sum_{n=1}^{\infty} (-1)^n \cdot \frac{1}{n}$$ 
the context is convergence and so on. And this series serves as an exampel for a not absolut convergent but convergent series. For me it looks like the harmonic series, which clearly diverges:
$$\sum_{n=1}^{\infty} \frac{(-1)^n}{n}$$=$$\sum_{n=1}^{\infty} |\frac{1}{n}|$$
Am I wrong?
 A: When we test for absolute convergence, you are correct, you have the harmonic series, which does not converge. Hence, your series is not absolutely convergent.
$$\sum_{n=1}^{\infty} \left|(-1)^n \cdot \frac{1}{n}\right| = \sum_{n = 1}^\infty \frac 1n$$
But your series is not equal to the harmonic series, since the terms alternate between positive and negative. Because of this, and for other reasons, while NOT absolutely convergent, $$\sum_{n=1}^{\infty} (-1)^n \cdot \frac{1}{n} = -1 + \frac 12 - \frac 13 + \frac 14 - \cdots $$ is convergent, and we can make this conclusion using the alternating series test.

When you have an alternating series, like this one, where the terms alternative between positive and negative


*

*First we check if a series is absolutely convergent. If so, we are done. Absolute convergence guarantees convergence. 

*If a series is not absolutely convergent, we must then check for
(non-absolute) convergence, using the alternating series test (also sometimes called Leibniz's Test).

A: $\frac{(-1)^n}n\ne\left|\frac1n\right|$ if $n$ is odd.
