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yuradex [85]
3 years ago
10

PLEASE HELP ME 40 PTS!!!!!!

Mathematics
1 answer:
SSSSS [86.1K]3 years ago
8 0

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

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the graph in the attached figure

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we have

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