sequences and series - If $n$ is a positive integer, Prove that $frac1{2^2}+frac1{3^2}+dotsb+frac1{n^2}ltfrac{2329}{3600}.$

If $n$ is a positive integer, Prove that

$$\frac1{2^2}+\frac1{3^2}+\dotsb+\frac1{n^2}\lt\frac{2329}{3600}.$$

please don't refer to the famous $1+\frac1{2^2}+\frac1{3^2}+\dotsb=\frac{\pi^2}6$.

I am looking a method that doesn't use $\text{“}\pi\text{''}.$

Unfortunately, I know and tried only $\text{“}\pi\text{''}$ method.