Can I split this into two series and apply the alternating series test?

Let's say we have a sequence $(a_n)$ such that $sgn(a_{n+2}) = -1 * sgn(a_{n})$ and the sequence has absolute value going to zero. I want to argue that the series converges conditionally as follows:

$\sum_{n=0}^{\infty}{a_n} = \sum_{n=0}^{\infty}{a_{2n}} + \sum_{n=0}^{\infty}{a_{2n + 1}}$

Each of the sums on the right side converge by the alternating series test. But, I'm not sure if this means that the left side converges because of fear that the series on the left may converge only conditionally and, as such this argument may fail because of rearrangements and/or some other issue. Does this work?

Please note: I now know this is not a "rearrangement" in the sense of having some bijection $\phi:\mathbb{N}\rightarrow\mathbb{N}$ and summing $\sum_{n=0}^{\infty}{a_\phi(n)}$. It is just the because such a rearrangement can take a conditionally convergent series and change its sum to anything that I'm also worried something similar can happen here.

Solution 1:

Yes, note that from the algebra of limits you have that if $B_n$ and $C_n$ are two convergent sequences then $B_n + C_n$ is also a convergent sequence and converges to the sum of the limits. So if we let $$ B_n = \sum_{2k \leq n}a_{2k}, \ C_n = \sum_{2k+1 \leq n}a_{2k+1} $$ then these are two convergent sequences. We can then write $$ A_n = \sum_{k \leq n} a_n = B_n + C_n. $$ We know that the right hand side converges and so the left hand side must also converge.