Giải thích các bước giải:

Ta có:

$\dfrac1{2^2}+\dfrac1{3^2}+\dfrac1{4^2}+\dfrac1{5^2}+\dfrac1{6^2}+\dfrac1{7^2}+\dfrac1{8^2}$

$<\dfrac1{1.2}+\dfrac1{2.3}+\dfrac1{3.4}+\dfrac1{4.5}+...+\dfrac1{7.8}$

$<1-\dfrac12+\dfrac12-\dfrac13+\dfrac13-\dfrac14+....+\dfrac17-\dfrac18$

$<1-\dfrac18$

$<1$

Ta có:

$\dfrac1{2^2}+\dfrac1{3^2}+\dfrac1{4^2}+\dfrac1{5^2}+\dfrac1{6^2}+\dfrac1{7^2}+\dfrac1{8^2}$

$>\dfrac1{2^2}+\dfrac1{3.4}+...+\dfrac1{8.9}$

$>\dfrac14+\dfrac13-\dfrac14+...+\dfrac18-\dfrac19$

$>\dfrac14+\dfrac13-\dfrac19$

$>\dfrac13+(\dfrac14-\dfrac19)$

$>\dfrac13$