# 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.