Show that there exist a constant c>0

Question

Show that there exist a constant $c>0$ such that for all $x \in [1,\infty)$ $$ \sum_{n>=x}^{\infty} 1/n^2 < c/x. $$

Answer 1

Note that for $x\ge 1$

$$\sum_{n>=x}^{\infty} 1/n^2 <\int _x^{\infty} \frac {1}{t^2}dt= \frac{1}{x} <2/x$$

Thus $c=2$ does the trick.

Answer 2

If $x$ is an integer bigger than $1$, you can bound the sum by

$$\int_{x-1}^\infty \frac{1}{ y^2 } \; d y.$$

If $x=1$, you have to tweak it a bit. Then if $x$ is not an integer, you have a different little tweak.