\(S=\dfrac{1}{4}+\dfrac{2}{4^2}+\dfrac{3}{4^3}+...+\dfrac{2013}{4^{2013}}+\dfrac{2014}{4^{2014}}\)
\(\Rightarrow4S=1+\dfrac{2}{4}+\dfrac{3}{4^2}+...+\dfrac{2013}{4^{2012}}+\dfrac{2014}{4^{2013}}\)
\(\Rightarrow4S-S=1+\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{2013}}-\dfrac{2014}{4^{2014}}\)
\(\Rightarrow3S=1+\dfrac{1}{4}+\dfrac{1}{4^2}+...+\dfrac{1}{4^{2013}}-\dfrac{2014}{4^{2014}}\)
\(\Rightarrow12S=4+1+\dfrac{1}{4}+...+\dfrac{1}{4^{2012}}-\dfrac{2014}{4^{2013}}\)
\(\Rightarrow12S-3S=4+\dfrac{2014}{4^{2014}}-\dfrac{2013}{4^{2013}}\)
\(\Rightarrow9S=4-\dfrac{1}{4^{2013}}.\dfrac{3019}{12}< 4\)
\(\Rightarrow S< \dfrac{4}{9}< \dfrac{1}{2}\)
