Evaluate the infinite geometric series: $$1-\frac{2}{7}+\frac{4}{49}-\frac{8}{343}+\dots$$
I recognize this as an infinite geometric series, where each term is obtained by multiplying the previous term by a constant ratio. To find the sum of such a series, I need to know the first term and the common ratio. The first term is 1, and the common ratio is -2/7, since that's what I multiply 1 by to get -2/7, and so on. Now, I recall that the formula for the sum of an infinite geometric series is $$S = \frac{a}{1-r},$$ where a is the first term and r is the common ratio, as long as the absolute value of r is less than 1. In this case, a = 1 and r = -2/7, and the absolute value of -2/7 is less than 1, so the formula applies. Plugging these values into the formula, I get $$S = \frac{1}{1-(-2/7)} = \frac{1}{1+2/7} = \frac{1}{9/7} = \frac{7}{9}.$$ 