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Using the binomial theorem as an approximation

  1. Oct 1, 2008 #1
    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. jcsd
  3. Oct 1, 2008 #2

    gabbagabbahey

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    The expansion is valid for all values of [itex]x[/itex], but only useful when [itex]x \ll 1[/itex]. If you let x=3.2, then successive terms in the expansion become larger and larger ([itex]3.2^3 >3.2^2[/itex]). But if [itex]x \ll 1[/itex], 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 [itex](4.2)^{-1/2}[/itex] as [itex]4^{-1/2}(1.05)^{-1/2}=(1/2)(1.05)^{-1/2} [/itex]; this way you can expand the square root about a much smaller x.
     
  4. Oct 1, 2008 #3

    HallsofIvy

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