+865 Evaluate the infinite geometric series: \(1-\frac{2}{7}+\frac{4}{49}-\frac{8}{343}+\dots\)
+310 Leave the 1 out for now. Combine -2/7 and 4/49. This equals -10/49. Combine -4/49 and 16/2401. This equals -40/2401.
This forms an infinite geometric series with first term -10/49 and common ratio 4/49. (You can go on a few more times to see that it is in fact a geometric series).
The sum is \(\frac{-10/49}{1-\frac{4}{49}}=\frac{-2}{9}\). We add this to our beginning 1 to find our final answer 1-2/9=7/9
