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Question about a tricky/difficult Taylor expansion of natural logarithm

  1. Mar 30, 2009 #1
    Can someone please tell me how to expand

    \ln(x + \sqrt{1+x^2})

    for small [itex]x[/itex]? I'd like to retain terms at least up to order [itex]x^5[/itex]. Thanks!
  2. jcsd
  3. Mar 31, 2009 #2
    Let f(x) = ln( x + sqrt(1 + x^2) ).

    Find f(0).
    Find f '(0)
    Find f ''(0)
    Find f '''(0)
    Find f ''''(0)
    Find f '''''(0)

    Then write f(x) = [f(0)x^0]/0! + [f '(0)x^1]/1! + [f ''(0)x^2]/2! + [f '''(0)x^3]/3! + [f ''''(0)x^4]/4! + [f '''''(0)x^5]/5!.

    Does that sound right?
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