Answer:
a) The given sequence is diverges
= ∞
Step-by-step explanation:
<u><em>Explanation</em></u>
Given that the
term sequence
![a_{n} = \frac{n^{3} - n}{n^{2} + 5 n}](https://tex.z-dn.net/?f=a_%7Bn%7D%20%3D%20%5Cfrac%7Bn%5E%7B3%7D%20-%20n%7D%7Bn%5E%7B2%7D%20%2B%205%20n%7D)
we have to prove that a given sequence converges or diverges
![\lim_{n \to \infty}a_{n} = \lim_{n \to \infty} \frac{n^{3} - n}{n^{2} + 5 n}](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7Da_%7Bn%7D%20%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7Bn%5E%7B3%7D%20-%20n%7D%7Bn%5E%7B2%7D%20%2B%205%20n%7D)
![= \lim_{n \to \infty} \frac{n^{3}(1- \frac{1}{n^{3} }) }{n^{2}(1+ \frac{5n}{n^{2} } )}](https://tex.z-dn.net/?f=%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7Bn%5E%7B3%7D%281-%20%5Cfrac%7B1%7D%7Bn%5E%7B3%7D%20%7D%29%20%7D%7Bn%5E%7B2%7D%281%2B%20%5Cfrac%7B5n%7D%7Bn%5E%7B2%7D%20%7D%20%29%7D)
![= \lim_{n \to \infty} \frac{n(1- \frac{1}{n^{3} }) }{(1+ \frac{5n}{n^{2} } )}](https://tex.z-dn.net/?f=%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%20%5Cfrac%7Bn%281-%20%5Cfrac%7B1%7D%7Bn%5E%7B3%7D%20%7D%29%20%7D%7B%281%2B%20%5Cfrac%7B5n%7D%7Bn%5E%7B2%7D%20%7D%20%29%7D)
= ∞
= ∞
The given sequence is diverges
18x-24+2x
20x-24
You simply multiply the 6 with everything in the fraction and the add the 2 x to it.