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?

- Aug 24th 2009, 05:08 PMserenaleeHow 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? - Aug 24th 2009, 05:19 PMKrizalid
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.