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Find the $n^{\text {th }}$ Taylor polynomial of $y=\ln x$ centered at $x=1$.
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We begin by creating a table of derivatives of $\ln x$ evaluated at $x=1$. While this is not as straightforward as it was in the previous example, a pattern does emerge, as shown below.
$\begin{array}{lll} f(x)=\ln x & \Rightarrow & f(1)=0 \\ f^{\prime}(x)=1 / x & \Rightarrow & f^{\prime}(1)=1 \\ f^{\prime \prime}(x)=-1 / x^2 & \Rightarrow & f^{\prime \prime}(1)=-1 \\ f^{\prime \prime \prime}(x)=2 / x^3 & \Rightarrow & f^{\prime \prime \prime}(1)=2 \\ f^{(4)}(x)=-6 / x^4 & \Rightarrow & f^{(4)}(1)=-6 \\ \vdots & & \vdots \\ f^{(n)}(x)= & \Rightarrow & f^{(n)}(1)= \\ \frac{(-1)^{n+1}(n-1) !}{x^n} & & (-1)^{n+1}(n-1) ! \end{array}$
we have,
\begin{aligned} p_n(x) &=f(c)+f^{\prime}(c)(x-c)+\frac{f^{\prime \prime}(c)}{2 !}(x-c)^2+\frac{f^{\prime \prime \prime}(c)}{3 !}(x-c)^3+\cdots+\frac{f^n(c)}{n !}(x-c)^n \\ &=0+(x-1)-\frac{1}{2}(x-1)^2+\frac{1}{3}(x-1)^3-\frac{1}{4}(x-1)^4+\cdots+\frac{(-1)^{n+1}}{n}(x-1)^n . \end{aligned}
Note how the coefficients of the $(x-1)$ terms turn out to be "nice."
by Platinum (141,884 points)

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