Check if sequence is bounded using the monotony

Question

I want to check if the following sequence is upper bounded, under bounded and bounded.

• $a_n=\frac{n^{30}}{2^n}$

We have that $\lim_{n \to +\infty} a_n=0$ and $a_1=\frac{1}{2}$.

So the sequence is not increasing.

The sequence is decreasing for the $n$ for which $a_{n+1}> a_n$.

This holds for $\left( 1+\frac{1}{n}\right)^{30} \geq 2$.

So we have to find the $n$ for which the last relation holds in order to find where the sequence is bounded, right?

How can we solve the inequality as for $n$?

Or can we see otherwise the lower bound of the sequence?

Answer 1

Let $\varphi(x)=\log(x^{30}/2^{x})$, then $\varphi'(x)=30/x-x\log 2<0$ if and only if $x>30/\log 2$, for $n\geq(\text{the least integer greater or equal to }30/\log 2$), the sequence is decreasing.