which convergence test is better?

Question

I need to check this one for absolute convergence $$\sum^{\infty}_1 \frac {(-1)^n(n+4)}{(n^2+1)^{1/4}(2+\sqrt{n^2+3})}$$

But I am not sure which method to use, it fails with Root or Ratio tests.

Answer 1

Since $\displaystyle\lim_{n\to\infty}\frac{n+4}{(n^2+1)^{\frac{1}{4}}(2+\sqrt{n^2+3})}\div\frac{1}{n^{\frac{1}{2}}}=\lim_{n\to\infty}\frac{n^{\frac{1}{2}}(n+4)}{n^{\frac{1}{2}}(1+\frac{1}{n^2})^{\frac{1}{4}}\left(2+n\sqrt{1+\frac{3}{n^2}}\right)}$

$\displaystyle=\lim_{n\to\infty}\frac{1+\frac{4}{n}}{(1+\frac{1}{n^2})^{\frac{1}{4}}\left(\frac{2}{n}+\sqrt{1+\frac{3}{n^2}}\right)}=1$ and $\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$ diverges,

the series of absolute values diverges by the Limit Comparison Test.