\(\frac{1}{x}+\frac{1}{2.x}+\frac{1}{6x}+\frac{1}{12x}+\frac{1}{30x}\)
= \(\frac{1}{x}\left(1+\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{30}\right)\)
= \(\frac{1}{x}\left(1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}\right)\)
= \(\frac{1}{x}\left(1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}\right)\)
=\(\frac{1}{x}\left(1+1-\frac{1}{6}\right)\)
=\(\frac{1}{x}.\frac{11}{6}\)
=\(\frac{11}{6x}\)