Finding the power series expansion of Ln (x) / x with respect to X-1

lnx=ln(x-1+1)=(x-1)-(x-1)^2/2+(x-1)^3/3-.
1/x=1/(1+x-1)=1-(x-1)+(x-1)^2-(x-1)^3+.
Let ln (x) / x = A0 + A1 (x-1) + A2 (x-1) ^ 2 + A3 (x-1) ^ 3 +
[(x-1)-(x-1)^2/2+(x-1)^3/3-.]
=[a0+a1(x-1)+a2(x-1)^2+a3(x-1)^3+.][1-(x-1)+(x-1)^2-(x-1)^3+.]
Comparison coefficient: A0 = 0, a1-a0 = 1, A2 + a0-a1 = - 1 / 2
ln(x)/x=(x-1)+(x-1)^2/2+.

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