Does $\frac{3}{1\cdot 2} - \frac{5}{2\cdot 3} + \frac{7}{3\cdot 4} - ...$ Converges? $$\frac{3}{1\cdot 2} - \frac{5}{2\cdot 3} + \frac{7}{3\cdot 4} - ...$$
Do you have an idea about this serie? If it converges what is the sum?

This is
$$\sum_{k=1}^{\infty} (-1)^{k+1} \frac{2 k+1}{k (k+1)}$$
For large $k$, the summand behaves as $2(-1)^k/k$.  By comparison to
$$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k} = \log{2}$$
the sum converges.
Note that the sum may be expressed as
$$\sum_{k=1}^{\infty} (-1)^{k+1} \left ( \frac{1}{k} + \frac{1}{k+1} \right ) = \log{2} + (1-\log{2}) = 1$$
