Study the convergence of $\sum_{n=1}^{\infty}\frac{n+\log{(n)}}{(n+\cos{(n)})^3}$ Study the behaviour of $$\sum_{n=1}^{\infty}\frac{n+\log{(n)}}{(n+\cos{(n)})^3}$$.
I have tried to use the comparison test considering that $\log{(n)}>1$ definitely and $1+\cos{(n)}<n+1$, then:
$$\frac{n+\log{(n)}}{(n+\cos{(n)})^3}>\frac{n+1}{(n+1)^3}=\frac{1}{(n+1)^2}$$
Now the series of $\frac{1}{(n+1)^2}$ converges but this does tell me anything about the convergence of the original series.
Can you help me in studying this series?
 A: Hint. Show that eventually the following estimates hold
$$\frac{n+\log(n)}{(n+\cos(n))^3}<\frac{n+n}{(n-1)^3}\leq\frac{3}{(n-1)^2}.$$
What may we conclude?
A: For $n\geq 2$ we have
$$\frac{n+\log{(n)}}{(n+\cos{(n)})^3}<\frac{n+n}{\left(n-\frac{n}{2}\right)^3}=\frac{2n}{\frac{n^3}{8}}=\frac{16}{n^2}$$
Thus
$$\sum_{n=1}^\infty\frac{n+\log{(n)}}{(n+\cos{(n)})^3}<\frac{1}{1+\cos(1)}+\sum_{n=2}^\infty \frac{16}{n^2}=\frac{1}{1+\cos(1)}+\frac{8 \pi ^2}{3}-16$$
A: Hint:
Use asymptotic equivalence:

*

*$\log n=_\infty o(n)$, so $n+\log n\sim_\infty n$;

*$\cos n$ is bounded, therefore $n+\cos n\sim_\infty n$.

Can you proceed from there?
A: Observation One
Informal Section of Observation One
For very large values of $n$, ${(n+\cos{(n)})^3}$ is aproximatly equal to $n^{3}$$\displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{(n+\cos{(n)})^3}$
Formal Section of Observation One
Special Theory of Observation One

$\forall \epsilon \in \left\{r \in \mathbb{R}: r > 0 \right\},$
  $\exists k \in \mathbb{N} :$
    $\forall n \in \mathbb{N},$
       $n \geq k \implies \biggl|(n+\cos{(n)})^{3} - n^{3} \biggr| < \epsilon$

General Theory of Observation One

$\forall F, G \subseteq \mathbb{N} \times \mathbb{R},$
  $\forall \epsilon \in \{r \in \mathbb{R}: r > 0\},$
    $\exists k \in \mathbb{N} : \forall n \in \mathbb{N}, n \geq k \implies \biggl|{F(n) - G(n)} \biggr| < \epsilon$

Observation Two
Informal & General Section of Observation Two

If function $F$ and function $G$ looks pretty similar to each-other as you go really really far to the right (along the x-axis), then $\displaystyle \sum_{n=1}^{\infty}H(F(n))$ converges if and only if $\displaystyle \sum_{n=1}^{\infty}H(G(n))$ converges

Formal and Special Section of Observation Two

If $\displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{(n+\cos{(n)})^3}$ converges then $\displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{(n+\cos{(n)})^3} = \displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{n^{3}}$

Formal and General Section of Observation Two

$\forall F, G, H \subseteq \mathbb{N} \times \mathbb{R},$
  If
     $\forall \epsilon \in \{r \in \mathbb{R}: r > 0\}, \exists k \in \mathbb{N} : \forall n \in \mathbb{N}, n \geq k \implies \biggl| {F(n) - G(n)}               \biggr| < \epsilon$
  then
    $\displaystyle \sum_{n=1}^{\infty}H(F(n))$ converges
      if and only if
    $\displaystyle \sum_{n=1}^{\infty}H(G(n))$ converges

Informal & Special Section of Section of Observation Two

$\displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{(n+\cos{(n)})^3}$ converges if and only if $\displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{n^{3}}$ converges

Observation 3

For very large values of $n, \quad n+\log{(n)} \approx n$.
$\displaystyle \sum_{n=1}^{\infty}\dfrac{n+\log{(n)}}{(n+\cos{(n)})^3}$ converges if and only if $\displaystyle \sum_{n=1}^{\infty}\dfrac{n}{n^{3}}$ converges.

