# Taylor polynomial...

• December 4th 2009, 06:51 AM
Unenlightened
Taylor polynomial...
Find the n-th degree Taylor polynomial $P_n(x)$ of $f(x)=\ln x$ at c=1. Calculate $P_3(1.1)$ and estimate the accuracy of the approximation $\ln 1.1$ $P_3(1.1)$.

Any suggestions?
• December 4th 2009, 07:40 AM
nehme007
The Taylor series expansion of f(x) about c = 1 is:
$f(x) = \sum_{n = 0}^{\infty} \frac{f^{(n)}(c)}{n!}(x-1)^n$.
This is also $P_n(x)$. For $f(x) = ln(x)$, $f(1) = 0$ and $f^{(n)}(1) = (-1)^{n+1} (n-1)!$. So:
$ln(x) = \sum_{n = 1}^{\infty} \frac{(-1)^n}{n}(x-1)^n$.
To calculate $P_3(1.1)$, 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).