Ta có:
\(S=\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{100}}\)
\(\Rightarrow7S=1+\frac{1}{7}+\frac{1}{7^2}+...+\frac{1}{7^{99}}\)
\(\Rightarrow7S-S=1-\frac{1}{7^{100}}\)
\(\Rightarrow6S=1-\frac{1}{7^{100}}\)
\(\Rightarrow S=\left(1-\frac{1}{7^{100}}\right)\div6=\frac{1}{6}-\frac{1}{7^{100}\times6}\)