Let x=1+a+a2+..... and y=1+b+b2.... where |a|<1 and |b|<1. Prove that 1+ab+a2b2+....=xyx+y−1
Using C,P, we get
x=1+a+a2+......
x=11−a
y=1+b+b2+......
y=11−b
and Let P=1+ab+a2b2+......
P=11−ab
and xyx+y−1=11−a×11−b11−a+11−b−1
xyx+y−1=11−b+1−a−1+a+b−ab=11−ab
Since P=11−ab=xyx+y−1
Hence Proved.