Question

Determine whether the sequence converges or diverges. if it converges, find the limit. (if an answer does not exist, enter dne. ) an = n2 n3 5n

Answer 1

The sequence $a_n=\frac{n^2}{n^3+5n}$ diverges.

For given question,

We have been given a sequence $a_n=\frac{n^2}{n^3+5n}$

We need to determine whether the sequence converges or diverges.

From given sequence we have, $a_{n+1}=\frac{(n+1)^2}{(n+1)^3+5(n+1)}$

We use Ratio Test.

According to Ratio Test, $r=\frac{a_{n+1}}{a_n}$, where sequence converges if and only if |r| < 1.

Consider,

$\lim_{n \to \infty} |\frac{a_{n+1}}{a_n} |\\\\= \lim_{n \to \infty} |\frac{\frac{(n+1)^2}{(n+1)^3+5(n+1)}}{\frac{n^2}{n^3+5n}} |\\\\\\= \lim_{n \to \infty} |\frac{(n+1)^2(n^3+5n)}{n^2[(n+1)^3+5(n+1)]} |\\\\=\infty$

Since $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n} |$ is not defined, the sequence $a_n=\frac{n^2}{n^3+5n}$ diverges.

Therefore, the sequence $a_n=\frac{n^2}{n^3+5n}$ diverges.

Learn more about the sequence here:

https://brainly.com/question/17175513

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