Project 8: Series

Instructions

Use Maxima to work this project. Each group should email an annotated Maxima file to the instructor.

Power Series

Consider the function \(f(x) = \ln ( 1 + x^7) \).

1. Use example 6 in the book to find a power series for \( \ln ( 1 + x^7) \)
2. Integrate the power series to approximate \( \displaystyle \int_0^{0.9} \ln ( 1 + x^7 ) \, dx \)
3. Use the integrate() command in Maxima to find \( \displaystyle \int_0^{0.9} \ln ( 1 + x^7 ) \, dx \). Explain why this isn't such a good idea (this is where the text cells ctrl-1 would come in handy).
4. Use the romberg() command in Maxima to approximate \( \displaystyle \int_0^{0.9} \ln ( 1 + x^7 ) \, dx \)

Taylor Series Approximation

This problem is to create a graph similar to figure 1 on page 478 of your textbook. Come up with a function (don't make it too simple) and pick a point in the domain of the function. Hint: Maclaurin series are easier to work with by hand, but the computer can work easily with other values.

1. Find the values of your function and the first three non-zero derivatives of your function at the point you picked.
2. Use the Taylor command in Maxima to verify your answers.
3. Graph the function and three approximations on the same graph. Zoom in tight enough that you can see the interesting portion and the difference in the approximations.