Use the Growth Rates of Sequences Theorem to find the limit of the following sequence or state that they diverge. $$ \left\{\frac{n^{20}}{\ln ^{40} n}\right\} $$

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We start with the sequence

$$
a_n = \frac{n^{20}}{\ln^{40} n}.
$$

As $$ n\to\infty $$, both $$ n^{20} $$ and $$ \ln^{40} n $$ tend to infinity. However, the Growth Rates of Sequences Theorem tells us that for sufficiently large $$ n $$, any positive power of $$ n $$ will grow faster than any positive power of $$ \ln n $$.

Specifically, for any positive numbers $$ a $$ and $$ b $$, we have

$$
\lim_{n\to\infty} \frac{n^a}{(\ln n)^b} = \infty.
$$

In our sequence, $$ a = 20 $$ and $$ b = 40 $$, so

$$
\lim_{n\to\infty} \frac{n^{20}}{\ln^{40} n} = \infty.
$$

Thus, the sequence diverges to infinity.
The sequence $$ \frac{n^{20}}{\ln^{40} n} $$ grows without bound as $$ n $$ increases, so it diverges to infinity.

Use the Growth Rates of Sequences Theorem to find the limit of the following sequence or state that they diverge. \( \left\{\frac{n^{20}}{\ln ^{40} n}\right\} \)

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