Consider two infinite geometric series. The first has leading term a, common ratio b, and sum S.The second has a leading term b, common ratio a, and sum \(\frac{1}{S}\). Find the value of a+b.
We have that
a / [ 1 - b] = S (1) and
b / [ 1 - a ] = 1/S we can write this as
[ 1 - a ] / b = S (2)
Equating (1) and (2) we have that
a / [ 1 - b ] = [ 1 - a ] / b cross-multiply
ab = [ 1 - b ] [ 1 - a ] simplify
ab = 1 - a - b + ab subtract ab from both sides
0 = 1 - (a + b) add (a + b) to both sides
(a + b ) = 1