If $\sum\limits_{n=0}^{\infty} a_n\ $ is a conditionally convergent series, then is $\sum\limits_{n=0}^{\infty} |a_n+a_{n+1}|\ $ convergent

Question

If $\sum\limits_{n=0}^{\infty} a_n\ $ is a conditionally convergent series, then is the sum of the average of consecutive terms necessarily convergent?

In other words, is $\sum\limits_{n=0}^{\infty} |a_n+a_{n+1}|\ $ convergent?

For the most standard example, the alternating harmonic series $\sum\limits_{n=0}^\infty {(-1)^n\over n+1},$ we get that $\sum\limits_{n=0}^{\infty} |a_n+a_{n+1}|\ $ is half the sum of the reciprocal triangular numbers, which does converge.

But does every conditionally convergent (but not necessarily alternating sign) series have this property, or is there one that diverges?

I feel like I'm missing some obvious triangle inequality trick, but I just don't see it.

Answer 1

No. Take your example, and interweave it with the all-zero sequence: $$ a_{n} = \begin{cases} \frac{(-1)^{n/2}}{n+1} &\text{ if } n \text{ even}\\ 0 &\text{ if } n \text{ odd} \end{cases} $$