Using the binomial theorem as an approximation

1. Oct 1, 2008

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.

2. Oct 1, 2008

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.

3. Oct 1, 2008

HallsofIvy

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