Check convergence $\sum\limits_{n=1}^\infty\left(\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}\right)$ Please help me to check convergence of $$\sum_{n=1}^{\infty}\left(\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}\right)$$

Update. Compare 
$$\sum_{n=1}^{\infty}\left(\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}\right)
$$
with the serie
$$
\sum_{n=1}^{\infty} \frac{1}{n^a}
$$
for a convenient $a\geq 1$.
Note that if $a_n=\left(\sqrt{1+\frac{7}{n^{2}}}-\sqrt[3]{1-\frac{8}{n^{2}}+\frac{1}{n^{3}}}\right)$,  $b_n=\frac{1}{n^a}$ and
$$
0<\lim_{n\to\infty} \frac{a_n}{b_n}<\infty.
$$
then $\sum_{n=1}^{\infty} a_n<\infty$ if, only if, $\sum_{n=1}^{\infty} b_n<\infty$.
