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.

Guest Feb 11, 2019

#1**+1 **

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

CPhill Feb 12, 2019