## Exercises

1. For the following functions and base points, determine what minimum order is required so that the Taylor polynomial approximates the function to within a tolerance of over the given interval.
1. , base point , interval .
2. , base point , interval .
3. , base point , interval .
4. , base point , interval .
5. ,base point , interval . (Hint: You may want to use axes=none at the end of this plot command to see what is happening in the plot.)
2. For the function, , use the TayPlot command to plot the function and multiple Taylor polynomial approximations of various orders with base point on the same graph over the interval ; use a y-range from to .
1. If you increase the order of the Taylor polynomial, can you get a good approximation at ?
2. Can you make a good guess at the radius of convergence of the Taylor series for ?
3. Repeat all of exercise 2 using and the same ranges.

A theorem from complex variables says that the radius of convergence of the Taylor series of a function like is the distance between the base point ( in this case) and the nearest singularity of the function. By singularity, what is meant is a value of where the function is undefined. Where is undefined? Is the distance between this point and the base point consistent with your guess of the radius of convergence from the plot?