Đáp án:
Giải thích các bước giải:
$\text{Đặt M = $\dfrac{1}{4^2}$ + $\dfrac{1}{5^2}$ + $\dfrac{1}{6^2}$ +...+ $\dfrac{1}{64^2}$}$
$\text{M < $\dfrac{1}{4^2}$ + $\dfrac{1}{4.5}$ + $\dfrac{1}{5.6}$ +...+ $\dfrac{1}{63.64}$}$
$\text{⇒ M < $\dfrac{1}{4^2}$ + $\dfrac{1}{4}$ - $\dfrac{1}{5}$ + $\dfrac{1}{5}$ - $\dfrac{1}{6}$ +...+ $\dfrac{1}{63}$ - $\dfrac{1}{64}$ +}$
$\text{⇒ M < $\dfrac{1}{4^2}$ + $\dfrac{1}{4}$ - $\dfrac{1}{64}$}$
$\text{⇒ M < $\dfrac{1}{4^2}$ + $\dfrac{16}{64}$ - $\dfrac{1}{64}$}$
$\text{⇒ M < $\dfrac{1}{4^2}$ + $\dfrac{15}{64}$}$
$\text{⇒ M < $\dfrac{1}{16}$ + $\dfrac{15}{64}$}$
$\text{⇒ M < $\dfrac{4}{64}$ + $\dfrac{15}{64}$}$
$\text{⇒ M < $\dfrac{19}{64}$ < $\dfrac{20}{64}$= $\dfrac{5}{16}$}$
$\text{Vậy M < $\dfrac{5}{16}$}$