Thread: How do i determine whether a series is convergent or divergent?

What's the systematic step that I can follow whenever they throw questions about series?
eg. sum from n=1 to infinity n(n+2) / (n+3)^2.
Should I find the nth term first? then calculate the limits? how?

The divergence test is one of the simplest methods to decide whether a series does converge or not.

In this case, it's quite simple: compute the limit of the general term, and you'll see that is not zero, hence, the series diverges.

Now, another proof could be bounding below the general term as follows so the series diverges (again) by direct comparison test with the Harmonic series.