<s>[INST] Series convergence or divergence . $\sum_{n=1}^\infty \left(\frac{n^4}{n^5+7}\right)$ $$\sum_{n=1}^\infty \left(\frac{n^4}{n^5+7}\right)$$
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 ?
 [/INST] 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.
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