Lời giải:
Gọi $S=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^n}$
$5S=1+\frac{1}{5}+....+\frac{1}{5^{n-1}}$
$\Rightarrow 4S=5S-S=1-\frac{1}{5^n}$
$S=\frac{1}{4}-\frac{1}{4.5^n}$
Khi $n\to \infty$ thì $4.5^n\to \infty\Rightarrow \frac{1}{4.5^n}\to 0$
$\Rightarrow \lim S = \frac{1}{4}-\lim \frac{1}{4.5^n}=\frac{1}{4}$