Question

Which statement describes the sequence defined by a Subscript n Baseline = StartFraction n cubed minus n Over n squared + 5 n EndFraction?

Answer 1

The answer is " The sequence converges to infinity. "

Explanation:

Given:

`\to a_n=(n^3-n)/(n^2+5n)`

`\lim_(n \to \infty) a_n= \lim_(n \to \infty) (n^3-n)/(n^2+5n)`

`= \lim_(n \to \infty) (n(n^2-1))/(n(n+5))\\\\= \lim_(n \to \infty) ((n^2-1))/((n+5))\\\\= \lim_(n \to \infty) (n(n-(1)/(n)))/(n(1+(5)/(n)))\\\\= \lim_(n \to \infty) ((n-(1)/(n)))/((1+(5)/(n)))\\\\`

Denominator
`= \lim_(n \to \infty) 1+(5)/(n)=1+\lim_(n\to \infty) (5)/(n)=1+0=1`

Numerator
`=\lim_(n\to \infty)n-(1)/(n)=\infty`

`\therefore\\\\\lim_(n\to \infty)a_n=(\infty)/(1) =\infty`