Finding limit of the sequence

Given a sequence defined as

$${a_{n}}= \frac{1} {\sqrt{2}} + \frac{2} {3} + \frac{3} {\sqrt{28}} + \cdots \cdots + \frac{n} {\sqrt{n^3+1}}$$

and I want to find the limit of this sequence.

I think the limit is positive infinity, but I don't know what I should do in order to prove this.

Please give me some hints on how to approach this, thanks to anybody who helps.

Answer

$$
\frac{n}{\sqrt{n^3+1}}\sim\frac1{\sqrt{n}},
$$
hence

$$
\lim_{n\to\infty}a_n=\infty.
$$

Explanation, using series' methods

$$
\lim_{n\to\infty}\frac{\dfrac{n}{\sqrt{n^3+1}}}{\dfrac1n}=1
$$
and the series

$$
\lim_{n\to\infty}\frac1{\sqrt{n}}
$$
is divergent, hence also
$$
\lim_{n\to\infty}\frac{n}{\sqrt{n^3+1}}
$$
id divergent, but $a_n$'s are partial sums of this series.

More elementary explanation

It is easy to see, that
$$
\dfrac{n}{\sqrt{n^3+1}}\geq\frac1n
$$
for all $n\geq2$. Let
$$
b_n=1+\frac12+\dots+\frac1n.
$$
It is certainly increasing, hence it remains to show, that it is unbounded. Let us consider

$$
b_{2^n}=1+\frac12+\left(\frac13+\frac14\right)+\left(\frac15+\dots+\frac18\right)+\dots
+\left( \frac1{2^{n-1}+1}+\dots+\frac1{2^n}\right).
$$
But the value in every parentheses is greater than $\frac12$, hence $$
b^{2^n}>1+\frac{n}{2}.
$$