Alternating Harmonic Series is Conditionally Convergent

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Theorem

The alternating harmonic series:

$\ds \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^\paren {n - 1} } n = 1 - \frac 1 2 + \frac 1 3 - \frac 1 4 + \cdots$

is conditionally convergent.


Proof

Note first that:

$\ds \sum_{n \mathop = 1}^\infty \size {\frac {\paren {-1}^\paren {n - 1} } n} = \sum_{n \mathop = 1}^\infty \frac 1 n$

which is divergent by Harmonic Series is Divergent.

Hence by definition $\ds \sum_{n \mathop = 1}^\infty \size {\frac {\paren {-1}^\paren {n - 1} } n}$ is not absolutely convergent.

$\blacksquare$


Next note that $\size {\dfrac 1 n}$ is a basic null sequence:

$\ds \lim_{n \mathop \to \infty} \dfrac 1 n = 0$

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.

$\blacksquare$


Also see


Sources

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