Taylor theorem and related
- Thread starter jkeller
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I would compute the first few derivatives:
\(\displaystyle f^{(1)}=\dfrac{1}{x-1}\)
\(\displaystyle f^{(2)}=-\dfrac{1}{(x-1)^2}\)
\(\displaystyle f^{(3)}=\dfrac{2}{(x-1)^3}\)
\(\displaystyle f^{(4)}=-\dfrac{6}{(x-1)^4}\)
Next, I would use induction, and state the hypothesis \(\displaystyle P_k\):
\(\displaystyle f^{(k)}=\dfrac{(-1)^{k-1}(k-1)!}{(x-1)^k}\)
Next, differentiate both sides to see if you can obtain \(\displaystyle P_{k+1}\).
\(\displaystyle f^{(1)}=\dfrac{1}{x-1}\)
\(\displaystyle f^{(2)}=-\dfrac{1}{(x-1)^2}\)
\(\displaystyle f^{(3)}=\dfrac{2}{(x-1)^3}\)
\(\displaystyle f^{(4)}=-\dfrac{6}{(x-1)^4}\)
Next, I would use induction, and state the hypothesis \(\displaystyle P_k\):
\(\displaystyle f^{(k)}=\dfrac{(-1)^{k-1}(k-1)!}{(x-1)^k}\)
Next, differentiate both sides to see if you can obtain \(\displaystyle P_{k+1}\).