jmc

Question

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 1/S. Find the value of a+b.

Accepted Answer

Writing S in terms of a and b, a1-b=S and b1-a = 1S. Thus, equating the second equation with the reciprocal of the first, 1S=1-ba=b1-a.Cross multiplying and simplifying, ab=(1-a)(1-b) and the result is a+b=1.