\(B=\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{500^2}\)
\(>\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{500.501}\)
\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{500}-\frac{1}{501}\)
\(=\frac{1}{4}-\frac{1}{501}>\frac{1}{4}-\frac{1}{20}=\frac{1}{5}\)
\(B=\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{500^2}\)
\(< \frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{499.500}\)
\(=\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{499}-\frac{1}{500}\)
\(=\frac{1}{3}-\frac{1}{500}< \frac{1}{3}\)
