If ∑| a_(n) | is convergent, is ∑ a_(n) also convergent?

Question

If ∑|
`a_(n)` | is convergent, is ∑
`a_(n)` also convergent?

Answer 1

Not necessarily. Consider the series
`\displaystyle\sum_(n=1)^\infty\frac{(-1)^n}n` and
`\displaystyle\sum_(n=1)^\infty\frac1n`. Here
`a_n=\frac1n`.

The first series converges by the alternating series test, which says
`\sum(-1)^na_n` converges if
`\left|(-1)^na_n\right|=|a_n|` is a decreasing sequence and converges to 0. This is the case, as
`a_n=\frac1n\to0` as
`n\to\infty`, and each term is decreasing. (Indeed the series converges to
`-\ln2`.)

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