Final answer:
The repeating decimal 0.7777... can be expressed as the infinite geometric series 0.7 + 0.07 + 0.007 + ..., which simplifies to the fraction 7/9.
Step-by-step explanation:
Expressing the repeating decimal 0.7777 as a geometric series can be done by recognizing that 0.7777... (with the 7s repeating indefinitely) is equal to 0.7 + 0.07 + 0.007 + 0.0007 + ..., and so on. Each term in this series is a common ratio (r) of 0.1 times the previous term. The first term (a) is 0.7. Therefore, this series is a geometric series, where each term can be represented as a * r^n, where n is the number of the term minus 1 (starting with n=0 for the first term).
To represent 0.7777... as an infinite geometric series, we use the formula for the sum of an infinite geometric series S = a / (1 - r), where |r| < 1. Plugging in our values, we get S = 0.7 / (1 - 0.1) = 0.7 / 0.9 = 7/9. Therefore, 0.7777... equals 7/9 when expressed as a fraction or the sum of an infinite geometric series.