Regarding Remainder of an Alternating Series

Alternating series test (Wikipedia.org)

Definition of an alternating series:
A series of the form $$\sum_{k=0}^\infty (-1)^k a_k$$
where all the $a_k$ are non-negative. The emphasis here is on the non-negative.

We know the absolute value of the remainder $R_n=\sum_{k=n+1}^\infty (-1)^k a_k$ for an alternating series will be less than the next term in the series, $a_{n+1}$.

So what's the point of all this? Well, some power series might look like alternating series, but they'll only be alternating series for certain values of $x$. Thus, you can only estimate the remainder using the above estimate for values of $x$ for which the power series turns into an alternating series.

[Remark: It sort of follows from the above, that a power series that doesn't look like an alternating series, might be an alternating series for certain values of $x$]