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?
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 $\displaystyle
\frac{n^{2}+2n}{n^{2}+6n+9}>\frac{2n}{16n^{2}}=\fr ac{1}{8n},$ so the series diverges (again) by direct comparison test with the Harmonic series.