Using the binomial theorem as an approximation

[SpaceAnimals8]

Use the binomial expansion of (1+x)^(-1/2) to find an approximation for 1/(rt4.2).

I've got the expansion of (1+x)^(-1/2) as 1-(1/2)x+(3/8)x^2...
but the obvious idea of substituting x=3.2 gives me the wrong answer. I think it's something to do with the expansion being valid but can't remember. Any help would be much appreciated.

 

[gabbagabbahey]

The expansion is valid for all values of x, but only useful when x \ll 1. If you let x=3.2, then successive terms in the expansion become larger and larger (3.2^3 >3.2^2). But if x \ll 1, then successive terms become smaller and smaller and so only the first few terms are large enough to make a significant contribution to the total value, and you can neglect higher order terms.

Try rewriting (4.2)^{-1/2} as 4^{-1/2}(1.05)^{-1/2}=(1/2)(1.05)^{-1/2}; this way you can expand the square root about a much smaller x.

 

[HallsofIvy]

You would be better off expanding (4+x)^1/2. Can you get the Binomial theorem expansion for that?