\( ( \frac { 1 } { 8 } ) + ( \frac { 1 } { 8 } ) ^ { 2 } + ( \frac { 1 } { 1 } ) ^ { 3 } + ( \frac { 1 } { 8 } ) ^ { 4 } + ( \frac { 1 } { 8 } ) ^ { 5 } + \ldots . \)

This series is an example of an infinite geometric series, which is a fascinating aspect of mathematics involving patterns and summation. The first term \(a\) is \(\frac{1}{8}\), and the common ratio \(r\) is also \(\frac{1}{8}\). The formula for the sum \(S\) of an infinite geometric series is given by \(S = \frac{a}{1 - r}\), provided that \(|r| < 1\). Plugging in the values, we see that this series converges to \(S = \frac{1/8}{1 - 1/8} = \frac{1/8}{7/8} = \frac{1}{7}\). Now, let's see how this can be put to use! Infinite geometric series pop up not only in math problems but also in real-life applications, such as calculating the present value of an annuity in finance. For example, if you expect to receive a fixed sum of money regularly and know the interest rate, you can use the formula for the sum of an infinite series to determine the worth of those future payments in today's dollars. Understanding these concepts can help you make smarter financial decisions!