I like to wade slowly into infinite series with the following two examples.
(1) Consider the following "proof" that 0=1.
\begin{eqnarray*}
0 &=& (1-1) \\
&=& (1-1) + (1-1) \\
&=& (1-1) + (1-1) + \dots \\
&=& 1 + (-1+1) + (-1+1) + \dots \\
&=& 1
\end{eqnarray*}
Students like this one because it feels like a party trick. But it's a useful illustration of the danger of handling infinite sums as if they were really long finite sums--assuming that every infinite series converges, and casually rearranging the order of summation--and will help you emphasize that infinite sums really can't work the same way that finite sums do.
(2) You can prove that $0.999\dots = 1$ with series as follows.
$$ 0.999\dots = \sum_{i=1}^{\infty} \frac{9}{10^i} = \sum_{i=1}^{\infty} \frac{10-1}{10^i} = \sum_{i=1}^{\infty} \left(\frac{1}{10^{i-1}} - \frac{1}{10^i}\right) = \frac{1}{10^0} = 1
$$
(You'll have to convince them that the last equality comes from infinitely many cancellations, but after example (1) they might think this is more of your numerical prestidigitation.) This example has a nice morale to it: that real numbers don't necessarily have unique decimal representations. It also gives students a taste for the kind arithmetic they'll be doing later on.
