# Example of a conditionally convergent series that is not alternating?

Every example of a conditionally convergent series I can think of is alternating. Can someone find a non-alternating conditionally convergent series? Thanks.

• Insert zeroes appropriately in any alternating series. – copper.hat Nov 7 '13 at 19:58

## 3 Answers

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 $\frac12\,\log2$.

Here is a non trivial example: $$\sum \frac{\sin n}{n}$$

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.

• Thanks. Another followup: is it true that all conditionally convergent series have an infinite number of negative terms? – r123454321 Nov 7 '13 at 20:03
• @RyanYu Yes. For if all but finitely many terms are positive, just replace finitely many terms with their absolute values. Changing finitely many terms does not affect convergence. – user61527 Nov 7 '13 at 20:04
• Yes. $\sum a_n=c$, $\sum |a_n|=\infty$. The difference is twice the sum of the negative terms, thus the sum of the negative terms is divergent, and thus there are infinitely many negative terms. – Tim Ratigan Nov 7 '13 at 20:06