Finding the sum of the series $\sum_{n=2}^{\infty} ((n^2+1)^{1/2} - (n^3+1)^{1/3})$ I need help in finding the sum of the series $\sum_{n=2}^{\infty} ((n^2+1)^{1/2} - (n^3+1)^{1/3})$ if it converges.
I can't even prove convergence. I tried comparison test. I tried telescoping or even sandwich the partial sums. But I could not get anything.
I also tried using the identity
$a-b = \frac{a^3-b^3}{a^2+ab+b^2}$
and even the identity for $a^6-b^6$
but it won't yield.
I found this one while going through some really old question papers of my college.
Please help.

Without series, we have
$$\sqrt{n^2+1}-\sqrt[3]{n^3+1} = \sqrt{n^2+1}- n + n - \sqrt[3]{n^3+1}$$
which allows us to use those conjugates without appealing to sixth order
$$= \frac{1}{n+\sqrt{n^2+1}} - \frac{1}{n^2+n\sqrt[3]{n^3+1}+(n^3+1)^{\frac{2}{3}}}$$
The divergence comes from the square root term.
