<s>[INST] 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?
 [/INST] $\frac{2n+1}{n(n+1)}=\frac{1}{n}+\frac{1}{n+1}$
So,
$\frac{3}{1.2}-\frac{5}{2.3} +\frac{7}{3.4}...= \frac{1}{1}+\frac{1}{2}-\frac{1}{2} -\frac{1}{3}+\frac{1}{3}+\frac{1}{4}....$
That should give you the idea.
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