Binomial Expansions!?

Question

Please help me on this HWK.

Show that if x is small,

(1+x) root (1-2x) approx = a + bx^2

Where a and b are constants to be stated.

I did the first part of this, which was to expand root(1-2x) given than mod(x) is less than 0.5, but no matter what i do i get an x^3 term!!!

Answer 1

there will be higher order terms.

but what they mean by "if x is small" .. is that you are only interested in the least small terms ... a + bx^2 ... and disregard higher order.

in my own notes I tend to write +O(x^3) or +o(x^2) to indicate that there are other terms of the order of x^3 or much smaller than x^2 (say I dont know whether next term is x^3 or x^4) as it helps to keep track of what's going on if the terms you thought would be important cancel out later on.