# Thread: Prove :

1. ## Prove :

Show that for any natural number the sum

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots + \frac{1}{n^2}$

lies between the values

$\left(1-\frac{2}{n+1}\right)\left(1-\frac{2}{2n+1}\right)\frac{\pi^2}{6}$ and $\left(1-\frac{1}{2n+1}\right)\left(1+\frac{1}{2n+1}\right) \frac{\pi^2}{6}.$

2. To start this, would one have to find the domain of the these two expressions over the natural numbers?

and

3. Originally Posted by newtoinequality
Show that for any natural number the sum

$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\ldots + \frac{1}{n^2}$

lies between the values

$\left(1-\frac{2}{n+1}\right)\left(1-\frac{2}{2n+1}\right)\frac{\pi^2}{6}$ and $\left(1-\frac{1}{2n+1}\right)\left(1+\frac{1}{2n+1}\right) \frac{\pi^2}{6}.$
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