Endless sums

Take a pie and prepare to cut it up. First divide it into two equal pieces, set one half aside, then cut the other half into two equal pieces again. Once more, set one piece aside and divide the other in two. Together, the pieces set aside represent the sum $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}=\frac{7}{8},\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{4\}+\backslash frac\{1\}\{8\}=\backslash frac\{7\}\{8\},$ while the remaining piece, which completes the pie, represents $\frac{1}{8}\backslash frac\{1\}\{8\}$.

Cutting without crumbs ---------------------

Suppose this pie produces no crumbs and can be divided for as long as we like, each time cutting the remaining piece into two equal parts and setting one aside.

After n divisions, the remaining piece will represent $\frac{1}{2^n}\backslash frac\{1\}\{2^n\}$ of the pie, which can be written as $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^n}=1-\frac{1}{2^n}\mathrm{.}\backslash frac\{1\}\{2\}+\backslash frac\{1\}\{4\}+\backslash frac\{1\}\{8\}+\backslash cdots+\backslash frac\{1\}\{2^n\}=1-\backslash frac\{1\}\{2^n\}.$