Expand (1-x)^(1/2) * (1+2x)^(1/2)
(1-x)^(1/2) * (1+2x)^(1/2)
= 1 + (1/2) x - (9/8)x^2 + (9/16)^3 - (117/128)x^4 + (279/256)x^5 + ...
the range of validity for the expansion of (1-x)^(1/2) is |x|<1
the range of validity for the expansion of (1+2x)^(1/2) is |x|< 1/2
taking the intersection, the expansion of (1-x)^(1/2) * (1+2x)^(1/2) is valid for |x|< 1/2.
Here come my questions,
(1-x)^(1/2) * (1+2x)^(1/2)
=(1+x-2x^2)^(1/2)
if I expand using Binomial theorem, the result is exactly the same.
However, when I solve |x-2x^2|<1 for the range of validity,
I got -0.414<x<1 or 1<x<2.414 (rejected as 1-x should be positive)
So -0.414<x<1!!! which is different from the previous |x|<1/2
You have solved |x-2x^2|<1 incorrectly.
May I ask why is the range of validity different? I thought since I am expanding the same expression so the range of validity should be unique, right?
Pls enlighten me