For the series $$\sum_{n=1}^{\infty}{\frac{n^2}{(n+1)(n^2+2)}},$$ application of the ratio test gave me a result of the limit equal to $1.$ This says that the test is inconclusive and should test for convergence using other ways. So I used the limit comparison test next. What I am not understanding is how did the answer in the book arrive at using $\sum\frac 1n$ as the comparison series. Can someone explain this for a calc 2 student. Please and thank you.
Convergence of series using ratio test.
sequences-and-series
Best Answer
When dealing with convergence of series, what really matters is the end behaviour of the series.
In this case, the $n$th term of the series is the rational function
$$f(n)=\frac{n^2}{(n+1)(n^2+2)}.$$
Therefore, as $n\to\infty,$ we see that $n+1$ becomes arbitrarily close to $n$ and $n^2+2$ becomes arbitrarily close to $n^2,$ so that $f(n)$ becomes arbitrarily close to $\frac{n^2}{n\cdot n^2}=\frac 1n.$ This ensures that the given series behaves like the harmonic series towards the tail.
This is called asymptotic comparison of functions, if you want to look further. But that's just the basic idea, as you asked.