a) Ta có: \(f\left(\frac{1}{3}\right)=\frac{1}{3}+\frac{1^2}{3^2}+\frac{1^3}{3^3}+....+\frac{1^{2016}}{3^{2016}}\)
\(\Rightarrow3.f\left(\frac{1}{3}\right)=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2015}}\)
\(\Rightarrow3.f\left(\frac{1}{3}\right)-f\left(\frac{1}{3}\right)=\left(1+\frac{1}{3}+...+\frac{1}{3^{2015}}\right)\)\(-\left(\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{2016}}\right)\)
\(\Rightarrow2.f\left(\frac{1}{3}\right)=1-\frac{1}{3^{2016}}\)
\(\Rightarrow f\left(\frac{1}{3}\right)=\frac{1-\frac{1}{3^{2016}}}{2}\)