Question

If ∑| x-formula"> | 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 $a_n=\dfrac1n$.

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 $a_n=\dfrac1n\to0$ 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.