Đặt A là tên biểu thức
\(A=1-\frac{15}{16}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{4n^2}\)
\(A=\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+...+\frac{1}{2^2n^2}\)
\(A=\frac{1}{2^2}\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{n^2}\right)\)
Ta có: \(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};\frac{1}{4^2}< \frac{1}{3.4};....;\frac{1}{n^2}< \frac{1}{\left(n-1\right)n}\)
\(A< \frac{1}{2^2}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{\left(n-1\right)n}\right)\)
\(A< \frac{1}{2^2}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{n-1}-\frac{1}{n}\right)\)
\(A< \frac{1}{2^2}\left(1-\frac{1}{n}\right)=\frac{1}{4}-\frac{1}{4n}< \frac{1}{4}\)(đpcm)