Why is $\frac{1}{3}=0.2\overline{31}$ in 5-adic expansion? 
Why is $\frac{1}{3}=0.2\overline{31}$ in 5-adic expansion ?

I get:
$\frac{1}{3}=\frac15+3\sum\limits_{k=1}^{\infty}\left(\frac15\right)^{2k}+\sum\limits_{k=1}^{\infty}\left(\frac15\right)^{2k+1}=0.13131313\dots=0.\overline{13}$
Something msut be wrong ?

What you've written is the base $5$ representation of the real number $\frac{1}{3}$.
The value $\frac{1}{3}$ in $5$-adic numbers is not just a different representation, but in some sense, a different number.
The standard $5$-adic representation of $\frac{1}{3}$ is actually $$\frac{1}{3}=2.\overline{31}.$$
Note the different placement of the decimal point from the expression in your question. $0.2\overline{31}$ is actually $\frac{5}{3}$ in $5$-adic numbers.
