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Sagot :
The sequence [tex]a_n=\frac{n^2}{n^3+5n}[/tex] diverges.
For given question,
We have been given a sequence [tex]a_n=\frac{n^2}{n^3+5n}[/tex]
We need to determine whether the sequence converges or diverges.
From given sequence we have, [tex]a_{n+1}=\frac{(n+1)^2}{(n+1)^3+5(n+1)}[/tex]
We use Ratio Test.
According to Ratio Test, [tex]r=\frac{a_{n+1}}{a_n}[/tex], where sequence converges if and only if |r| < 1.
Consider,
[tex]\lim_{n \to \infty} |\frac{a_{n+1}}{a_n} |\\\\= \lim_{n \to \infty} |\frac{\frac{(n+1)^2}{(n+1)^3+5(n+1)}}{\frac{n^2}{n^3+5n}} |\\\\\\= \lim_{n \to \infty} |\frac{(n+1)^2(n^3+5n)}{n^2[(n+1)^3+5(n+1)]} |\\\\=\infty[/tex]
Since [tex]\lim_{n \to \infty} |\frac{a_{n+1}}{a_n} |[/tex] is not defined, the sequence [tex]a_n=\frac{n^2}{n^3+5n}[/tex] diverges.
Therefore, the sequence [tex]a_n=\frac{n^2}{n^3+5n}[/tex] diverges.
Learn more about the sequence here:
https://brainly.com/question/17175513
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