convergent series and divergent series

Hi I have two questions.

First, $\sum_{n=1}^\infty \frac{n}{n^3+1}$.

Is it divergent or convergent? I think it seems like it is positive and decreasing function so we can apply integral test. however, integrating this function is not an easy task.

Is there any other test i can use?

Second, $\sum_{n=1}^\infty \frac{n^2+1}{n^3+1}$.

I can't think of any test,,

Can anyone give me an idea please~

• Don't the terms in the first sum resemble $1/n^2$? And in the second sum, don't they resemble $1/n$? – zhw. Apr 17 '15 at 3:34
• So for the first one I can use comparison test so that 1/n^2 is greater than that and is convergent and it must be converegent – Nancy Apr 17 '15 at 3:37
• for the second part I think I could use comparison test but can't see which one shall I use – Nancy Apr 17 '15 at 3:37

A great starting point on these questions is to "ignore all the small pieces" and see what happens. For instance, for large $n$, the $+1$ in the denominators won't really matter. So the first really looks like $$\sum_{n \geq 1} \frac{1}{n^2}$$ and the second looks like $$\sum_{n \geq 1} \frac{1}{n}.$$ Some calculus books have a so-called "limit comparison" test that makes this analysis rigorous. Barring that, let's see if we can do better.
In the first, $$\sum_{n \geq 1} \frac{n}{n^3 + 1} < \sum_{n \geq 1} \frac{n}{n^3} = \sum_{n \geq 1} \frac{1}{n^2}.$$
In the second, $$\sum_{n \geq 1} \frac{n^2 + 1}{n^3 + 1} > \sum_{n \geq 1} \frac{n^2}{n^3 + n^3} = \frac{1}{2}\sum_{n \geq 1} \frac{1}{n}.$$