If ∑| | is convergent, is ∑ also convergent?

Not necessarily. Consider the series and . Here .

The first series converges by the alternating series test, which says converges if is a decreasing sequence and converges to 0. This is the case, as as , and each term is decreasing. (Indeed the series converges to .)

On the other hand, the second series is a classic example of a divergent sum.