1/3+2/3^2+1/3^3+2/3^4+...
=1/3+1/3^3+1/3^5+...+2/3^2+2/3^4+2/3^6+...
=(1/3)(1+1/3^2+1/3^4+...)+(2/9)(1+1/3^2+1/3^4+...)
=(5/9)(1+1/9^1+1/9^2+...)
=lim(5/9)(1-1/9^n)/(1-1/9) (n→+∞)
=(5/9)/(8/9)
=5/8
=0.625