Alternating Harmonic Series is Conditionally Convergent

Theorem
The alternating harmonic series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac {\left({-1}\right)^\left({n - 1}\right)} n = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \cdots$

is conditionally convergent.

Proof
Note first that:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \left\vert{\frac {\left({-1}\right)^\left({n - 1}\right)} n}\right\vert = \sum_{n \mathop = 1}^\infty \frac 1 n$

which is divergent by [Harmonic Series is Divergent.

Next note that $\left\langle{\dfrac 1 n}\right\rangle$ is a basic null sequence:
 * $\displaystyle\lim_{n \to \infty} \dfrac 1 n = 0$

and that:
 * $\forall n \in \N_{>0}: \dfrac 1 n > 0$

From Reciprocal Sequence is Strictly Decreasing:


 * $\dfrac 1 n > \dfrac 1 {n+1}$

The result follows from the Alternating Series Test.

Also see

 * Alternating Harmonic Series sums to ln 2