Every example of a conditionally convergent series I can think of is alternating. Can someone find a non-alternating conditionally convergent series? Thanks.
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$\begingroup$Any convergent reordering of a conditionally convergent series will be conditionally convergent. A typical example is the reordering $$ 1,-\frac12,-\frac14,\frac13,-\frac16,-\frac18,\frac15,-\frac1{10},-\frac1{12},\frac17,-\frac1{14},\ldots $$ of the alternating harmonic series, with sum $\frac12\,\log2$.
$\endgroup$ $\begingroup$Here is a non trivial example: $$ \sum \frac{\sin n}{n} $$
$\endgroup$ $\begingroup$Trivial example:
$$1 + 0 - \frac{1}{2} + 0 + \frac{1}{3} + 0 - \frac{1}{4} + 0 + \dots$$
is a non-alternating version of the alternating harmonic series.
Not-as-trivial example: Consider a sequence of the form
$$\frac{1}{2}, \frac{1}{2}, -1,$$$$ \frac{1}{3}, \frac{1}{3} -\frac{2}{3},$$ $$\frac{1}{4}, \frac{1}{4}, -\frac{1}{2},$$
and so on. The series associated to this sequence converges (to $0$, in fact), but it's not absolutely convergent.
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