Project 8: Series


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

Power Series

1. Integrating a power series

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

  1. Use the Maclaurin series for \( \ln (1+x) \) given in Table 1of section 8.7 and a substitution 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 \( \displaystyle \int_0^{0.9} \ln ( 1 + x^7 ) \, dx \)

Note that Maxima uses log(x) instead of ln(x).

Taylor Series Approximation

2. Effect of increasing order on interval of convergence

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.

Differential Equations

3. Finding a power series solution to a differential equation

One application of a power series is to approximate the solution to a differential equation.

Let \( y = \displaystyle \sum_{n=0}^{\infty} c_n x^n \).

  1. Find \(y'\). Remember that the \(n = 0\) term is a constant, so it will disappear when you find the derivative. This means that you should change the index of summation when you write \(y\)'.
  2. Substitute \(y\) and \(y'\) into the equation \( (x^2 + 4)y' + y = e^x \).
  3. Use Maxima to find the first 6 non-zero terms in the solution.

Note that the example in the power series solutions document is harder than the one you're asked to work.

You should also be able to combine the left hand side into a single summation without using Maxima; that might be a question on the exam.