Leibniz test is wrong? Maybe I am reading the mathese wrong but according to my book: If $a_n$ is positive and decreasing and $\,\displaystyle\lim_{n\to\infty} a_n = 0,\,$ then the alternating series converges.
So for example if I have $\,\displaystyle\sum_1^\infty \frac{1}{n},\,$ I know that diverges so $\,\displaystyle\sum_1^\infty \frac{(-1)^n}{n}\,$ should also diverge but according to the Leibniz test it is decreasing and it's limit is zero so it should converge. What does this mean?
 A: The alternating series $$\sum_{n = 1}^\infty \frac{(-1)^n}{n}$$ converges, as you note, conditionally, and by the Leiniz test. It does not converge absolutely. Please note that 
$$\sum_{n = 1}^\infty \frac{(-1)^n}{n} \neq \sum_{n = 1}^\infty \frac 1n$$
Alternating series are not equivalent to their non-alternating counterparts. The alternating series test, aka the Leibniz Test, is precisely for series like this: series that do not converge absolutely, but converge nonetheless. The fact that alternating terms are negative negates the divergence of the sum in the case of the absolute term.
Absolute convergence simply means the series consisting of the absolute values of the terms converges. It does not mean "definitely" converges.  A conditionally convergent series still very much converges, for a different reason, but converges just the same.
A: Everything in the question makes sense EXCEPT "so $\sum_1^\infty \frac{(-1)^n}{n}$ should also diverge".  That's wrong.  It is NOT TRUE that if $\sum_{n=1}^\infty a_n$ converges, then so does $\sum_{n=1}^\infty |a_n|$.
