Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge? Is my solution correct?

Question

Does $\sum_1^\infty \frac{n}{n^2 + 4}$ converge or diverge?

I am confused because my friend insists the series converges conditionally. I think the series diverges. Here is my process and solution:

$\sum_1^\infty \frac{n}{n^2 + 4}$

Limit Comparison Test:

Let $\sum_1^\infty \frac{n}{n^2 + 4} = \sum_1^\infty a_n$

Let $\sum_1^\infty b_n = \sum_1^\infty \frac{1}{n}$

Since $$\lim \limits_{n \to \infty} | \frac{a_n}{b_n}| = \lim \limits_{n \to \infty} | \frac{\frac{n}{n^2 + 4}}{\frac{1}{n}}| = \lim \limits_{n \to \infty} \frac{n^2}{n^2 + 4} = \lim \limits_{n \to \infty} \frac{n^2}{n^2(1 + \frac{4}{n^2})} = \lim \limits_{n \to \infty} \frac{1}{1 + \frac{4}{n^2}} = 1$$

and $1 > 0$ and $1$ is a finite number,

we can say that the behavior of $\sum_1^\infty a_n$ is the same as the behavior of $\sum_1^\infty b_n$.

Since $\sum_1^\infty b_n$ diverges ( $\sum_1^\infty \frac{1}{n}$ diverges by p-series, $p=1$), $\sum_1^\infty a_n$ diverges.

So by the Limit Comparison Test, $\sum_1^\infty \frac{n}{n^2 + 4}$ diverges.

Right?

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Also, this problem was an exercise in the Ratio/Root Test section. Both tests, however seem to fail. Can the convergence be solved with the Root Test or the Ratio Test?

Answer 1

Direct comparison is a little easier here: $$\sum_{n=1}^\infty \frac{n}{n^2+4} \geq \sum_{n=2}^\infty \frac{n}{n^2+4} \geq \sum_{n=2}^\infty \frac{n}{n^2+n^2} = \frac{1}{2}\sum_{n=2}^\infty \frac{1}{n} = \infty$$

Answer 2

Write $\frac{n}{n^2 + 4} = \frac1{n + \frac{4}{n}}$. Note that $ n + \frac{4}{n} \le n + 4 $. Use comparison test.

Answer 3

Since your $a_n \geqslant \frac {1} {2n}$ except at $n = 1$ and $\sum \frac {1} {2n}$ diverges, you establish that $\sum a_n$ diverges.