How to show that the sequence is not monotonic

Question

Let the sequence $a_n=\frac{n^{30}}{2^n}$.

I want to check if the sequence is monotonic or strictly monotonic. If it is not monotonic, I want to check if it is monotonic from an index and on. Also if it is bounded.

I have thought to consider that the sequence is increasing.

Then

$$a_{n+1} \geq a_n \Rightarrow \frac{(n+1)^{30}}{2^{n+1}} \geq \frac{n^{30}}{2^n} \Rightarrow 2^n (n+1)^{30} \geq n^{30} 2^{n+1} \Rightarrow (n+1)^{30} \geq 2n^{30} \Rightarrow \left( \frac{n+1}{n}\right)^{30} \geq 2 \Rightarrow \left( 1+\frac{1}{n}\right)^{30} \geq 2$$

Can we find from this a restriction for $n$ and then conclude that the sequence is not increasing? And after that the same to show that $a_n$ is not decreasing?

Is there also an other way to show that the sequence is not monotonic?

Answer 1

Since $a_2>a_1$, the sequence is not decreasing. And since $a_1=\frac12$ and $\lim_{n\to\infty}a_n=0$, the sequence is not increasing.

Answer 2

Take $f(x)=\frac{x^{30}}{2^x}$ on the interval $[1,+\infty)$

$$f'(x)=\frac{30x^{29}2^x-x^{30}2^x\ln{2}}{4^x}$$

For $x>\frac{30}{\ln{2}}$ we have that $f'(x)<0$ so $f$ is decreasing on $(\frac{30}{\ln{2}},+\infty)$ and increasing on $[1,\frac{30}{\ln{2}}]$

Thus the sequence is strictly decreasing for every $n \geq [\frac{30}{\ln{2}}]+1$

So the sequence is not monotonic in general but we can say that it is ''eventually'' monotonic