$M=1+\frac{1}{3}-$ $\frac{1}{3^2}+...-$ $\frac{1}{3^{20}}$
=> $3M=3+1-\frac{1}{3}+...-$ $\frac{1}{3^{19}}$
=> $M+3M=(1+3+1)+($$\frac{1}{3}-$ $\frac{1}{3})+...-$ $\frac{1}{3^{20}}$
=> $4M=5-$$\frac{1}{3^{20}}$
=> $M=\frac{5}{4}-$ $\frac{1}{3^{20}.4}$
=> M=$\frac{5.3^{20}-1}{3^{20}.4}$