# Given series converges or diverges?

Given:

$$\sum_{n=4}^\infty \left(\frac {n^{2.5}-1}{n^{6.5}+4}\right)$$

I know that a given series converges if $lim: a_n \to 0$, in the above case for ${n \to \infty}$ the limit is 0. Hence, it should converge. Am I right?

No, $\lim a_n=0$ is not sufficient for a series to converge. However, for large $n$ the series above is like $\sum 1/n^4$ and it converges because $\sum 1/n^A$ converges for all $A>1$.
• The series can be compared to $\sum 1/n^4$ in the comparison test, I think. I'm not sure what a P series is. – ForgotALot Mar 6 '16 at 17:24
It is not true that a series converges if $\lim a_n=0$. For example, consider $$\sum_{n=1}^\infty\frac 1n.$$
Clearly, $\lim_{n\to\infty}\frac 1n=0$, but it is easily shown by the integral test that the series diverges.