Question

PLEASE HELP ME 40 PTS!!!!!!
Which statement describes the sequence defined by a Subscript n Baseline = StartFraction n cubed minus n Over n squared + 5 n EndFraction?

The sequence diverges.
The sequence converges to 0.
The sequence converges to ∞.
The sequence converges to –∞.

Answer 1

Answer:

The sequence converges to  infinity.

Step-by-step explanation:

$a_n = \frac{n^3 - n}{n^2 + 5n}\\\\ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n^3 - n}{n^2 + 5n}$

                $= \lim_{n \to \infty} \frac{n(n^2 - 1)}{n(n + 5)} \\\\= \lim_{n \to \infty} \frac{(n^2 - 1)}{(n + 5)}\\\\= \lim_{n \to \infty} \frac{n(n - \frac{1}{n})}{n(1 + \frac{5}{n})}\\\\= \lim_{n \to \infty} \frac{(n - \frac{1}{n})}{(1 + \frac{5}{n})}$

$denominator = \lim_{n \to \infty} 1 + \frac{5}{n} = 1 + \lim_{n \to \infty} \frac{5}{n} = 1+0 = 1$

$numerator = \lim_{n \to \infty} n - \frac{1}{n} = \infty$

Therefore,

             $\lim_{n \to \infty} a_n = \frac{ \infty}{1} = \infty$

Answer 2

The sequence diverges because the series divergence the limit tends to infinity does not exist option (A) is correct.

What is the convergent of a series?

A series is convergent if the series of its partial sums approaches a limit; that really is, when the values are added one after the other in the order defined by the numbers, the partial sums get closer and closer to a certain number.

We have:

A sequence:

$\rm a_n=\dfrac{\left(n^3-n\right)}{n^2+5n}$

As we know if the series converges the limit tends to infinity exist.

If the series divergence the limit tends to infinity does not exist.

Taking limit x tends to infinity:

$\rm = \lim \:_{n\to \:\infty \:\:}\dfrac{\left n^3-n\right}{n^2+5n}$

$\rm =\lim _{n\to \infty \:}\left(\dfrac{n^2-1}{n+5}\right)$

$\rm =\lim _{n\to \infty \:}\left(\dfrac{n-\frac{1}{n}}{1+\dfrac{5}{n}}\right)$

$\rm =\dfrac{\lim _{n\to \infty \:}\left(n-\dfrac{1}{n}\right)}{\lim _{n\to \infty \:}\left(1+\dfrac{5}{n}\right)}$

= ∞/1

= ∞  = does not exist.

Thus, the sequence diverges because the series divergence the limit tends to infinity does not exist option (A) is correct.

Learn more about the convergent of a series here:

brainly.com/question/15415793

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