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

*

*One where there is a sequence of only negative terms whose absolute value are decreasing, or;

*One where there is a sequence of only positive terms whose absolute value are decreasing, or;

*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
 A: 
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, $k_m=\max K_m$.
