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Question 3

3. (i) State the general formula for the Maclaurin series of a function $f(x)$ .

(ii) Prove by induction the formula

{{d^{n}}\over{dx^{n}}}\log(1+x)={{(-1)^{n-1}(n-1)!}\over{(1+x)^{n}}},

for $-1<x<1$ and all integers $n=1,2,\dots$ .

(iii) Hence or otherwise obtain the Maclaurin series for $f(x)=\log(1+x)$ .

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