Giải thích các bước giải:
Ta có :
$A=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+..+\dfrac{1}{50^2}$
$\to A=1+\dfrac{1}{2.2}+\dfrac{1}{3.3}+..+\dfrac{1}{50.50}$
$\to A<1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+..+\dfrac{1}{49.50}$
$\to A<1+\dfrac{2-1}{1.2}+\dfrac{3-2}{2.3}+..+\dfrac{50-49}{49.50}$
$\to A<1+\dfrac 11-\dfrac 12+\dfrac 12-\dfrac 13+..+\dfrac{1}{49}-\dfrac{1}{50}$
$\to A<1+1-\dfrac 1{50}$
$\to A<2-\dfrac 1{50}<2$