\(S=\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{63}\)
\(>\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+...+\frac{1}{8}\right)+\left(\frac{1}{9}+...+\frac{1}{16}\right)\)
\(>\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+...+\frac{1}{8}\right)+\left(\frac{1}{16}+...+\frac{1}{16}\right)\)
\(=\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\frac{1}{2}=2\)