\(M=1+\frac{1}{3}-\frac{1}{3^2}+\frac{1}{3^3}-\frac{1}{3^4}+......+\frac{1}{3^{19}}-\frac{1}{3^{20}}\)
\(\Rightarrow\frac{1}{3}M=\frac{1}{3}+\frac{1}{3^2}-\frac{1}{3^3}+\frac{1}{3^4}-\frac{1}{3^5}+.......+\frac{1}{3^{20}}-\frac{1}{3^{21}}\)
\(\Rightarrow\frac{1}{3}M+M=1+\frac{1}{3}+\frac{1}{3}-\frac{1}{3^{21}}\)
\(\Rightarrow\frac{4}{3}M=\frac{5}{3}-\frac{1}{3^{21}}\)\(\Rightarrow M=\frac{\frac{5}{3}-\frac{1}{3^{31}}}{\frac{4}{3}}\)