Does the series $\sum (1+n^2)^{-1/4}$ converge or diverge? The integral is $\int\left(\,1 + n^{2}\,\right)^{-1/4}\,{\rm d}n$ is not quite possible, so I should make a comparison test.
What is your suggestion?
EDIT:
And what about the series
$$
\sum\left(\, 1 + n^{2}\,\right)^{-1/4} \cos\left(\, n\pi \over 6\,\right)
$$
Does it converge or diverge?

For the second part, notice that there is a pattern to the factor of $\cos\left(\frac{\pi n}{6}\right)$ which repeats every 12 terms. Use this pattern to think of the sequence as alternating.
