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Taylor series of f(x)=ln(x+1) centred at 2

  1. May 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Taylor series of f(x)=ln(x+1) centred at 2

    2. Relevant equations

    from 0 to infinity ∑ cn(x-a)n

    cn = f(n)(a)/n!

    3. The attempt at a solution

    f(x) = ln(1+x)
    f'(x) = 1/(1+x)
    f''(x) = -1/(1+x)2
    f'''(x) = 2/(1+x)3
    f''''(x) = -6/(1+x)4


    f(2) = ln(3) = 1.0986
    f'(2) = 1/3
    f''(2) = -1/9
    f'''(2) = 2/27
    f''''(2) = -6/81

    f(2)+f'(2)(x-2)+f''(2)(x-2)2/2!+f'''(2)(x-2)3/3!+f''''(2)(x-2)4/4!

    I cant see any pattern except partially ..(-1)n-1(x-2)n/3nn!

    I have no idea what to do with ln3

    Please help!
     
  2. jcsd
  3. May 20, 2012 #2
    Just write the ln3 outside of the sum. Then try to think of a pattern for the remaining terms. First, start by trying to find a general formula for [itex]f^{(n)}(2)[/itex].
     
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