x=1+a+a+a^2+a^3+...... and y=1+b+b^2+b^3..... a&b are proper fraction 1+ab+a^2.b^2+......=?

Question

x=1+a+a+a^2+a^3+...... and y=1+b+b^2+b^3..... a&b are proper fraction 1+ab+a^2.b^2+......=?

Answer 1

Really a rather convoluted expression of a geometric series:

Since a and b are a proper fractions, both x and y converge

Therefore the sum shall converge with a common ratio ab.

 

S = a / (1 - r)

S = 1 / (1 - ab)