Answer
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=0$
(please see step-by-step)
Work Step by Step
Our inference, based on the graph (see below)
is that the sequence converges to 0.
Analytically, we use Th.9.1:
[ Let $L$ be a real number.
Let $f$ be a function of a real variable such that $\displaystyle \lim_{x\rightarrow\infty}f(x)=L$.
If $\{a_{n}\}$ is a sequence such that $f(n)=a_{n}$ for every positive integer $n$,
then $\displaystyle \lim_{n\rightarrow\infty}a_{n}=L$. ]
We define $f(x)=\displaystyle \frac{1}{x^{3/2}}$, for which $a_{n}=f(n)$ .
As $ x\rightarrow\infty$, the numerator becomes very large, and $f(x)\rightarrow 0.$
So,
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=\lim_{n\rightarrow\infty}\frac{1}{n^{3/2}}=\lim_{x\rightarrow\infty}\frac{1}{x^{3/2}}=0$.
