CC MATHEMATICS CHAOS AND CALCULUS SYLLABUS ASSIGNMENTS

Series Homework

Work on this assignment with a partner. Hand in one write-up for the two of you with well-written and clear explanations of your findings by Friday at 3PM.

1. Go to the website http://math.rice.edu/~lanius/Lessons/Series/infinite.htm. Answer the question posed and explain your reasoning (using the pictures shown here, even if you remember a different argument you learned before.) Can you think of picture like this one that would demonstrate the sum of any other infinite series? By this I mean a picture that shows EXACTLY what some infinite sum of fractions would have to equal.

2. Go to the website http://www.shout.net/~mathman/html/prob1.html. This page is written to introduce elementary school children to infinite series. It has a different graphic for visualizing the sum of a series, and asks you to generalize your answer to a general geometric series (question 3). Answer the first three questions posed here. In answering question 2, include both a colored in grid and a graph of the partial sums showing what they approach. Try to design your grid so that it shows EXACTLY what the sum has to be. (I don't think their answer (which you can get by following the link) is very good at doing this.) Do both a grid and a graph of partial sums for question 3 too. Try to find a formula for the partial sums. If finding a formula or a convincing grid picture is difficult in general, try the special case A=1. If you need more grid paper, you can find it at http://id.mind.net/~zona/mmts/graphPaper/simpleGraphPaper/graphPaper1.html

3. Explain how the formula you came up with in #2 relates to the formula for the sum of a geometric series given on page 395 of your calculus text.

4. In your Calculus text, do problems p398 #12 and 13.

5. Go to the website http://www.math.utah.edu/~carlson/teaching/calculus/harmonic.html and do the exercise suggested there. Follow the link to the page on Nicholas Oresme to get a hint for how to prove your answer. This same argument appears on page 396 of your calculus book. If you have trouble with the explanation on the website, look it up on page 396 and then explain it in your own words.

6. Give two examples of infinite series that diverge that have not already appeared in this assignment. One should have the property that the individual terms go to 0, and one should have the property that they do not.