[tex]x(1+ \frac{1}{2}+ \frac{1}{3}+...+ \frac{1}{2014})=2014-(1-\frac{1}{2}) -(1 -\frac{1}{3})-...-(1- \frac{1}{2014}) \\ \\x(1+ \frac{1}{2}+ \frac{1}{3}+...+ \frac{1}{2014})=2014-1+ \frac{1}{2}-1+ \frac{1}{3}-...-1+ \frac{1}{2014} \\ \\ x(1+ \frac{1}{2}+ \frac{1}{3}+...+ \frac{1}{2014}) =2014-2013+ \frac{1}{2} + \frac{1}{3}+...+ \frac{1}{2014} \\ \\x(1+ \frac{1}{2}+ \frac{1}{3}+...+ \frac{1}{2014})=1+ \frac{1}{2}+ \frac{1}{3}+...+ \frac{1}{2014} \Rightarrow \boxed{x=1} [/tex]
[tex]b)~x=1+3+3^2+3^3+...+3^{2013}= \frac{3^{2014}-1}{2} . \\ \\ \sqrt{2x+1}=\sqrt{2 \cdot \frac{3^{2014}-1}{2}+1 }= \sqrt{3^{2014}-1+1}= \sqrt{3^{2014}} =3^{1007}.[/tex]
