This is probably a simple question, but I just wanted to be sure.
Say, we have a finite alternating sum: $$S_m=\sum_{n=1}^m (-1)^{n+1} a_n$$
Where the terms are decreasing in absolute value: $$0<a_n<a_{n-1}$$
For an even number of terms it's easy to show that for any integers $q<p$:
$$S_{2q}<S_{2p}< S_{2q+1}$$
But for an odd number of terms we will also have to consider the last term:
$$S_{2q}+a_{2p+1}<S_{2p+1}< S_{2q+1}+a_{2p+1}$$
Is the above correct?
For $p > q$ you have $$ S_{2p} - S_{2q} = \sum_{j=q+1}^p (a_{2j-1} - a_{2j}) > 0 $$ and $$ S_{2p+1} - S_{2q+1} = \sum_{j=q+1}^p (-a_{2j} + a_{2j+1}) < 0 $$ and therefore $$ S_{2q} < S_{2p} < S_{2p+1} < S_{2q+1} \, . $$
Your estimate $S_{2q}+a_{2p+1}<S_{2p+1}$ is equivalent to $S_{2q+1}<S_{2p+1}$ and therefore wrong.
The other estimate $S_{2p+1}< S_{2q+1}+a_{2p+1}$ is correct, but holds even without the additional term $a_{2p+1}$ on the right-hand side.