\(B=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{25^2}\)
\(B=\left(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}\right)+\left(\dfrac{1}{4^2}+...+\dfrac{1}{25^2}\right)\)
\(B=\dfrac{49}{36}+\left(\dfrac{1}{4^2}+...+\dfrac{1}{25^2}\right)\)
\(B=\dfrac{1}{36}+\dfrac{4}{3}+\left(\dfrac{1}{4^2}+...+\dfrac{1}{25^2}\right)\)
\(B>\dfrac{4}{3}\left(1\right)\)
\(\)\(B< 1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{24.25}\)
\(B< 1+1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{24}-\dfrac{1}{25}\)
\(B< 2-\dfrac{1}{25}\)
\(B< 2\left(2\right)\)
Từ \(\left(1\right)\) và \(\left(2\right)\) ta có:
\(\dfrac{4}{3}< B< 2\)
\(\rightarrowđpcm\)
