Series that converges while the sequence doesn't necessarily decrease This might be a kind of stupid question but I can't seem to get my mind around it so I hope someone can help me.

There's this statement that says: 


  "If $a_n &gt; 0$ for every $n\in\Bbb N$, the series
  $\sum\limits_{n=1}^\infty(-1)^n\cdot a_n$ converges if and only if
  $a_n$ is a decreasing sequence".


I know this is false but I can't seem to find any example that contradicts it.

Thank you in advance to everyone.
Consider the nondecreasing sequence:

$$1, 2, 1, \frac{1}{2}, 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{2}, \frac{1}{4}, \ldots$$

It's easiest to see the pattern and the convergence if you look at the terms 3 at a time.
First, your title, "series that converges while the sequence doesn't" is impossible.  One of the first things one should learn about series is that "a necessary (but not sufficient) condition that the series $\sum a_n$ converges is that the sequence $\{a_n\}$ converges to 0.  That is not a "sufficient condition" because there exist examples of sequences that converge to 0 while the series does not converge $\sum\frac{1}{n}$ is such an example.  However, for alternating sequences, $\sum (-1)^n a_n$ it is both necessary and sufficient that $a_n$ converge to 0.