If we can't change the order of the terms of a conditionally convergent series, but can change the sign of terms, can we get whatever number we want?

Question

Let $\ \displaystyle\sum_{n\in\mathbb{N}} x_n\ $ be a conditionally convergent series, and we can choose $\ a_j=-1\ $ or $\ a_j=1\ $ for each $j\in\mathbb{N}.$

Let $\ \beta\in\mathbb{R}.$

Can we choose the $\ a_j\ $ so that

$$\sum_{n\in\mathbb{N}} a_n x_n = \beta\quad ? $$

If so, we can solve this question I came across this morning by letting $\ x_n = \left(-1\right)^n\frac{1}{p_n},\ $ where $\ p_n\ $ is the $\ n-$th prime number, because this is a conditionally convergent series. Edit: this is not what the question was asking.

Note that my question is different to the Riemann Series Theorem, which says that we can keep the terms, but rearrange the order. In my question we are changing the terms.

Answer 1

Proof outline

$\sum x_n\ $ is conditionally convergent implies:

$$\sum_{n\in\mathbb{N}} x_n =\gamma\in\mathbb{R}\qquad (1)$$

$$\sum_{n\in\mathbb{N}} \vert x_n \vert = \infty \qquad (2)$$

$$\ \vert x_n \vert \to 0^+\qquad (3)$$

$(1)\implies\ $ there exists $\ N\in\mathbb{N}\ $ such that $\ \displaystyle\sum_{i=1}^{k} x_i\ $ is very close to $\ \gamma\ $ for all $\ k\geq N.$

Next, we can use $\ (2)\ $ and $\ (3)\ $ to get from $\ \gamma\ $ to very close to $\ \beta.$