prove the given inequality (for series ) For any given $n \in \Bbb N,$ prove that, $$1+{1\over 2^3}+\cdots+{1\over n^3} <{3\over 2}.$$

Hint: Notice that as $f(x)=\frac{1}{x^3}$ is decreasing for every $x>0$ then the following expression holds
$$\sum_{k=1}^n \dfrac{1}{k^3}<\int_0^n\dfrac{1}{x^3}d x$$
