Tell if a sum is convergent $\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$ $$\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$$
I tried to solve this by saying that $$\frac{2}{n(n+1)} = \frac{2}{n} - \frac{2}{n+1}$$
I then made two sums like this:
$$\sum\limits_{n=1}^\infty \frac{2}{n} - \sum\limits_{n=1}^\infty \frac{2}{n+1}$$
And since we know that $1/n$ is divergent I said that $\sum\limits_{n=1}^\infty \frac{2}{n(n+1)}$ would also be divergent, which for some reason was wrong. So my question is simply what am I doing wrong?

Here are the steps,
$$ \sum\limits_{n=1}^\infty \frac{2}{n(n+1)} $$
$$= 2\sum\limits_{n=1}^\infty \left(\frac{1}{n}-\frac{1}{n+1}\right) $$
$$=2\lim\limits_{m\to\infty}\sum\limits_{n=1}^m \left(\frac{1}{n}-\frac{1}{n+1}\right) $$
$$=2\lim\limits_{m\to\infty} \left(1-\frac12+\frac12-\frac13 + \cdots +\frac{1}{m}-\frac{1}{m+1}\right) $$
$$=2\lim\limits_{m\to\infty} \left(1-\frac{1}{m+1}\right) $$
$$= 2(1-0) = 2$$
Therefore, this series converges to $2$.
