i have the following serie
$$ \sum_{k=1}^{\infty} \frac{{\sqrt{k} + k^3}}{{k^4+k^2}} $$
is it enough to calculate the limit to prove that it converges ?
so that would be
$$\lim_{k \to \infty } \frac{{\sqrt{k} + k^3}}{{k^4+k^2}} = 0 $$
so we may say the serie converges to 0.
doing it this way is wrong or right ?
This series actually does not converge. $$\sum_{k=1}^{\infty} \frac{{\sqrt{k} + k^3}}{{k^4+k^2}}$$ Let $$S_n=\sum_{k=1}^{n} \frac{{\sqrt{k} + k^3}}{{k^4+k^2}}$$
Then $S_n>\sum_{k=1}^{n} \frac{{k^3}}{{k^4+k^2}} =\sum_{k=1}^{n} \frac{k}{{k^2+1}}$ which is equivalent to the harmonic series $\sum_{k=1}^{n} \frac{1}{{k}}$, which diverges.
Note that the fact that the term of the series converges to zero does not guarantee at all the convergence of the series ; witness the harmonic series. To investigate whether the series converges or not, you have a number of different tools : D'Alembert test, Cauchy test, equivalences, majoration etc.