+4609 Evaluate the infinite geometric series: \(1-\frac{2}{7}+\frac{4}{49}-\frac{8}{343}+\dots\)
+128474 We can split this into two series......
The first is
1 + 4/49 +16/2401 + 64/ 117649 ...........
And the sum of this series = 1 / [ 1 - (2/7)^2 ] = 49/45
The second series can be written as
- [ 2/7, 8/343, 32/16807........]
And the sum of this series = - [ [2/7] / [1 - (2/7)^2 ] ] = -14/45
So.......the sum is (49/45) - (14/45) = 35/45 = 7/9
