Dễ thấy M>0.
Ta cần chứng minh M<1.Thật vậy!
\(M=\frac{1}{2^2}+\frac{1}{3^2}+.....+\frac{1}{45^2}\)
\(< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+....+\frac{1}{44\cdot45}\)
\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{44}-\frac{1}{45}\)
\(=\frac{1}{1}-\frac{1}{45}\)
\(< 1\)
\(\Rightarrow0< M< 1\)
\(\Rightarrowđpcm\)