Question

Use the Growth Rates of Sequences Theorem to find the limit of the following sequence or state that they diverge. {eq}{n^{16}/(\ln n)^{32}} {/eq}
Select the correct choice below and, if necessary, fill in the answer box to complete the choice
A. The limit of the sequence is___________ . (Simplify your answer.)
B. The sequence diverges____________.

Answer 1

Answer:

The sequence diverges ( B )

Explanation:

$\frac{n^{16} }{(In n)^{32} }$

Applying the Growth rates of sequences theorem to find the limit of the given sequence above

$\lim_{n \to \infty} \frac{n^{16} }{(In n)^{32} }$  = ∞   this means that

The sequence is divergent because the rate at which n increase is very much higher than the rate at which (In n) increases