To Quickly Find the Sum of a List of Fractions $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \frac { 1 } { 32 } + \frac { 1 } { 64 } + \frac { 1 } { 128 }$

Essay

To quickly find the sum of a list of fractions, such as $\frac { 1 } { 2 } + \frac { 1 } { 4 } + \frac { 1 } { 8 } + \frac { 1 } { 16 } + \frac { 1 } { 32 } + \frac { 1 } { 64 } + \frac { 1 } { 128 }$ mathematicians use the formula $S = \frac { b \left( 1 - r ^ { k } \right) } { 1 - r }$ Solve the formula for b .