2) No, the series sum is not the same as the sum of the absolute values.
If you can prove that the sum of the absolute values CONVERGES, then so does the alternating series. In fact, so does any series with the same absolute values (it doesn't even necessarily have to be alternating).
It's relatively easy to understand using the comparison test.
It should be obvious that any number is no greater than its absolute value.
Therefore, the sum of numbers is never any greater than the sum of their absolute values.
It should make sense that if the "larger" series converges to a number, than anything "smaller" must also converge to a (smaller) number.
So that means, if is convergent, then so is . If you prove convergence this way, then you say the series is ABSOLUTELY CONVERGENT.
Of course, if you prove that the sum of the absolute values is divergent, that doesn't mean that the original series is divergent. You would have to use some other test (e.g. Root Test, Lagranges Alternating Series Test). If a series is not absolutely convergent, but can still be proven to be convergent, then you say it is CONDITIONALLY CONVERGENT.