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 $a+b.$
I know that the sum of an infinite geometric series is given by the formula $S = a/(1-b),$ where $a$ is the leading term and $b$ is the common ratio. So, for the first series, I have $S = a/(1-b),$ and for the second series, I have $1/S = b/(1-a).$ I can cross-multiply the second equation to get $S = (1-a)/b,$ and then equate the two expressions for $S$ to get $a/(1-b) = (1-a)/b.$ I can cross-multiply again to get $ab = (1-a)(1-b),$ which I can expand to get $ab = 1 - a - b + ab.$ Simplifying, I get $a + b = 1.$ 