# 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 \(f(x) = \ln ( 1 + x^7) \).

- Use example 6 in the book to find a power series for \( \ln ( 1 + x^7) \)
- Integrate the power series to approximate \( \displaystyle \int_0^{0.9} \ln ( 1 + x^7 ) \, dx \)
- 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).
- 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*.

- Find the values of your function and the first three non-zero derivatives of your function at the point you picked.
- Use the Taylor command in Maxima to verify your answers.
- 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.