\( 1 + ( \frac { 2 } { 5 } ) + ( \frac { 2 } { 5 } ) ^ { 2 } + ( \frac { 2 } { 5 } ) ^ { 3 } + ( \frac { 2 } { 5 } ) ^ { 4 } + \cdots \)

The given expression is a geometric series with the first term \(a = 1\) and the common ratio \(r = \frac{2}{5}\). The sum of an infinite geometric series can be calculated using the formula: \[ S = \frac{a}{1 - r} \] where \(a\) is the first term and \(r\) is the common ratio. Substitute the values of \(a = 1\) and \(r = \frac{2}{5}\) into the formula to find the sum of the series. Calculate the value by following steps: - step0: Calculate: \(\frac{1}{\left(1-\frac{2}{5}\right)}\) - step1: Remove the parentheses: \(\frac{1}{1-\frac{2}{5}}\) - step2: Subtract the numbers: \(\frac{1}{\frac{3}{5}}\) - step3: Multiply by the reciprocal: \(\frac{5}{3}\) The sum of the given geometric series is \( \frac{5}{3} \) or \( 1\frac{2}{3} \) or \( 1.\dot{6} \).