Series convergence or divergence . $\sum_{n=1}^\infty \left(\frac{n^4}{n^5+7}\right)$

Question

$$\sum_{n=1}^\infty \left(\frac{n^4}{n^5+7}\right)$$

I try this series by the comparison test with $a_n\le b_n,$ $a_n=\frac{n^4}{n^5+7}$ and $b_n=\frac{n^4}{n^5}=\frac{1}{n}$

then $b_n$ diverges, dose the series diverges ?

Answer 1

A bit more formally:

Let $n \ge 2:$

$a_n=\dfrac{n^4}{n^5+7} \gt \dfrac{n^4}{n^5+n^5} =\dfrac{1}{2n}.$

Since $(1/2) \sum \dfrac{1}{n}$ diverges,

$\sum \dfrac{n^4}{n^5+7}$ diverges (comparison test).

Answer 2

$$ \frac{n^4}{n^5+7}\underset{(+\infty)}{\sim}\frac{1}{n} $$

What can you say about $\displaystyle \sum_{n \geq 1}^{ }\frac{1}{n}$?

Answer 3

As an alternative to the limit comparison test note that by binomial series

$$\frac{n^4}{n^5+7}=\frac{1}{n+\frac7{n^4}}=\frac1n\left(1+\frac7{n^5}\right)^{-1}\ge \frac1n\left(1-\frac7{n^5}\right)=\frac1n-\frac7{n^6}$$

thus $\sum_{n=1}^{\infty} \left(\frac{n^4}{n^5+7}\right)$ diverges.

Answer 4

Hint: $$\frac{n^4}{n^5+7}\geq\frac{n^4}{8n^5}=\frac{1}{8n}$$ for all $n\geq1$.