demonstration (follow along)
consider the IVP
y'' + y = 0, y(0) = 1, y'(0) = 0.
here the coefficients for the series satisfy (n+2)(n+1)a_{n+2} + a_n = 0,
i.e., a_n = -a_{n-2}/(n*(n-1)).
look at approximation up to t^10 term:
For[a = List[1, 0]; f = 1;
i = 2, i <= 10, i++, a = Append[a, -a[[i - 1]]/(i*(i - 1))];
f = f + a[[i+1]]*t^i];
p1 = Plot[f, {t, 0, 10}, PlotRange -> {{0, 10}, {-2, 2}}]
now compare with approximations using up to t^20, t^50.
do on your own
- use series to approximate solutions to the IVP
y'' - y = 0, y(0) = 1, y'(0) = 0.
here the coefficients for the series satisfy (n+2)(n+1)a_{n+2} - a_n = 0.
how many terms do you need to take to determine the shape of the solution?
- use series to approximate solutions to the IVP
y'' - y = 0, y(0) = 0, y'(0) = 1.
here the coefficients for the series satisfy (n+2)(n+1)a_{n+2} - a_n = 0.
how many terms do you need to take to determine the shape of the solution?
- use series to approximate solutions to the IVP
y'' + y' + t y = 0, y(0) = 1, y'(0) = 0.
how many terms do you need to take to determine the shape of the solution?
- use series to approximate solutions to the IVP
y'' + y' + t y = 0, y(0) = 0, y'(0) = 1.
how many terms do you need to take to determine the shape of the solution?
extra: convergence issues
Remember that series approximations are only good in the region of convergence.
Suppose you want to make a Taylor series for f(t) = log t. First,
the function is not defined at t=0, so you cannot expand around t=0.
If you expand around t=x for some positive x, then the radius of convergence
can only be x. Here is code to generate Taylor approximations to log t
about x. Try varying the values of x and see what happens.
For[a = List[0]; x = 1; f = Log[x];
i = 1, i <= 10, i++, a = Append[a, (-1)^(i + 1)/(i*x^i)];
f = f + a[[i + 1]]*(t - x)^i];
Plot[f, {t, 0, 10}, PlotRange -> {{0, 10}, {-5, 5}}]
course evaluations
after i leave a few minutes early, please the complete course evaluations.
while the university looks at the numerical responses,
for me personally, the most useful feedback are the written comments.
here are the instructions i received:
- click on the following link: http://eval.ou.edu.
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Type your OUNet ID (4+4) and your password into the login window and click Log In. This is the same login information that you would use to check your OU email or log into oZone.
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After your login information has been authenticated, you will see a list of all the classes in which you are enrolled this Spring.
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Click the Evaluate Now link next to each course to evaluate it.
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When you are finished evaluating a course, click Submit Evaluation at the bottom of the evaluation form to save it.
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You will then be returned to the course list page. From here you can evaluate another course or log out.
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