TITLE: How to prove that $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$
QUESTION [5 upvotes]: I have this sequence:
 $$\sum_{1}^{\infty} \frac{1}{n^3}$$
and I need to prove that:  $\sum_{1}^{\infty} \frac{1}{n^3} \le 1.5$
So basically I know that this sequence converges using the integral test, but I don't know how to prove the above statement.
Some help?

REPLY [4 votes]: The simplest is a comparison with a lower Riemann sum:
$$\sum_{n=2}^\infty\frac1{n^3}\le\int_1^{\infty}\frac{\mathrm d\mkern1mux}{x^3}=\biggl[-\frac1{2x^2}\biggr]_1^{\infty}=\frac12.$$