Can (the partial sums of) a conditionally convergent series always be written as an alternating sequence of decreasing terms?

Question

True or false:

If $\ \sum a_n\ $ is conditionally convergent series, then there exists an increasing sequence of integers $\ k_1,\ k_2,\ k_3,\ldots\ $ such that

$$\ \left(\ \displaystyle\sum_{n=1}^{k_1} a_n\ ,\ \sum_{n=k_1+1}^{k_2} a_n\ ,\ \sum_{n=k_2+1}^{k_3} a_n\ ,\ \ldots\right)\ $$

is an alternating sequence of decreasing terms.

Obviously, a conditionally convergent series must have an infinite amount of positive terms and an infinite amount of negative terms.

Edit: I think for any conditionally convergent, such a sequence of partial sums (split like above) should exist:

1. One where there is a sequence of only negative terms whose absolute value are decreasing, or;
2. One where there is a sequence of only positive terms whose absolute value are decreasing, or;
3. An alternating sequence of decreasing terms (as in the original question).

But I am not sure even of this. But this is the line of thinking I am going down.

Also, this question and it's sentiments are probably closely related to the Riemann Series Theorem, whose proof I admittedly do not know the details to.

If the answer is yes, then this would help me answer some other, more specific questions I have on conditionally convergent series (because it would be a useful way to re-write the series without rearranging the terms.)

Further edit: Vaguely related question

Answer 1

True or false:

If $\ \sum a_n\ $ is conditionally convergent series, then there exists an increasing sequence of integers $\ k_1,\ k_2,\ k_3,\ldots\ $ such that

$$\left(\ \displaystyle\sum_{n=1}^{k_1} a_n\ ,\ \sum_{n=k_1+1}^{k_2} a_n\ ,\ \sum_{n=k_2+1}^{k_3} a_n\ ,\ \ldots\right)$$

is an alternating sequence of decreasing terms.

True.

Partition the positive integers as follows:

Let $a_1\in K_1$, then, for any $m,n\in\Bbb Z^+$:

If $a_n\in K_m$ and $a_n\cdot a_{n+1}\ge0$, then $a_{n+1}\in K_m$.

If $a_n\in K_m$ and $a_n\cdot a_{n+1}<0$ then $a_{n+1}\in K_{m+1}$.

It follows that if $\displaystyle\sum_{n\in K_m}a_n$ is positive, then $\displaystyle\sum_{n\in K_{m+1}}a_n$ is negative, and vice-versa. That the sequence...

$$A_m=\sum_{n\in K_m} a_n$$

...decreases (in absolute value, which is what I assume is meant) is guaranteed by the fact $A_m$ must converge to zero in order that $\sum_m A_m$ be convergent.

We in turn know that $\sum_m A_m$ is converges, because it is exactly equal $\sum_n a_n$.

Finally, $k_m=\max K_m$.