Find the n-th degree Taylor polynomial $\displaystyle P_n(x)$ of $\displaystyle f(x)=\ln x$ at c=1. Calculate $\displaystyle P_3(1.1)$ and estimate the accuracy of the approximation $\displaystyle \ln 1.1$≈$\displaystyle P_3(1.1)$.
Any suggestions?
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Find the n-th degree Taylor polynomial $\displaystyle P_n(x)$ of $\displaystyle f(x)=\ln x$ at c=1. Calculate $\displaystyle P_3(1.1)$ and estimate the accuracy of the approximation $\displaystyle \ln 1.1$≈$\displaystyle P_3(1.1)$.
Any suggestions?
The Taylor series expansion of f(x) about c = 1 is:
$\displaystyle f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-1)^n $.
This is also $\displaystyle P_n(x)$. For $\displaystyle f(x) = ln(x) $, $\displaystyle f(1) = 0 $ and $\displaystyle f^{(n)}(1) = (-1)^{n+1} (n-1)! $. So:
$\displaystyle ln(x) = \sum_{n = 1}^{\infty} \frac{(-1)^n}{n}(x-1)^n $.
To calculate $\displaystyle P_3(1.1)$, let x = 1.1 and add up the terms in the series up to n = 3. You should get something close to ln(1.1).