Convergent Series: $\sqrt{n+1}/(n^2 + 1)$

Question

I have this homework problem that I'm having difficulty on.

$$ \sum_{n=1}^\infty \frac{\sqrt{n+1}}{n^2+2} $$

I've found that the series is most likely convergent by the divergent test. As using lhospitals rule, the sequence approaches 0, thus it MIGHT be a convergent series. However, I do think it's convergent because it doesn't appear to be a harmonic sequence. So now I need to find the sum of the series, and that's where I am having trouble with.

Answer 1

Hint: Compute the limit$$\lim_{n\to\infty}\frac{\dfrac{\sqrt{n+1}}{n^2+1}}{\dfrac1{n^{3/2}}}.$$

Answer 2

Note that for $n\ge1$, we have the estimate

$$0<\frac{\sqrt{n+1}}{n^2+2}\le \frac{\sqrt{n+n}}{n^2}=\frac{\sqrt2}{n^{3/2}} $$

Inasmuch as the series $\sum_{n\ge 1}\frac1{n^{3/2}}$ converges, the series of interest, $\sum_{n\ge1 }\frac{\sqrt{n+1}}{n^2+2}$ does likewise.