University of Florida/Egm4313/s12.team11.gooding/R4

Problem Statement
Find n sufficiently high so that $$y_n(x_1), y'_n(x_1)$$ do not differ from the numerical solution by more than $$10^{-5}$$ at $$x_1=0.9$$

Solution
Using a program in MATLAB that iteratively added terms onto the taylor series of $$log(1+x)$$, terms were added until the error between the exact answer and the series was less than $$ 10^{-5}$$.



It was found after trial and error that $$ n=39 $$ for the error to be of a magnitude of $$ 10^{-5}$$. This error found was 9.7422e-005

Similarly, for $$y'_n(x_1)$$.



It was found after trial and error that $$ n=74 $$ for the error to be of a magnitude of $$ 10^{-5}$$. This error found was 9.3967e-005

Problem Statement
Develop $$log(1+x)$$ in Taylor series about $$\hat{x}=1$$ for $$ n=4,7,11$$ and plot these truncated series vs. the exact function. What is now the domain of convergence by observation?

Solution
A MATLAB program was created, which calculated the Taylor series of each n value, along with the exact function, then plotted these together to show the comparison of all the series. Below is the Taylor series for $$n=7$$ expanded at $$\hat{x}=1$$. $$\frac{x - 1}{2\, \ln\!\left(10\right)} - \frac{{\left(x - 1\right)}^2}{8\, \ln\!\left(10\right)} + \frac{{\left(x - 1\right)}^3}{24\, \ln\!\left(10\right)} - \frac{{\left(x - 1\right)}^4}{64\, \ln\!\left(10\right)} + \frac{{\left(x - 1\right)}^5}{160\, \ln\!\left(10\right)} - \frac{{\left(x - 1\right)}^6}{384\, \ln\!\left(10\right)} + \frac{\ln\!\left(2\right)}{\ln\!\left(10\right)} $$

It can be seen by observation that the domain of convergence has shifted to the right one unit. --egm4313.s12.team11.gooding (talk) 03:48, 14 March 2012 (UTC)