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This test consists of 5 multiple choice questions, 5 short answer questions, and 10 short essay questions.

Multiple Choice Questions

1. What did Cantor define as the continuum?
(a) The square root of any real number.
(b) All imaginary numbers.
(c) All imaginary and real numbers.
(d) Real numbers between 0 and 1.

2. In the Bernoulli's time, what was the current definition of a series?
(a) The sum of a finite series of terms.
(b) The infinite sum of a convergent series.
(c) The finite sum of a divergent series.
(d) The sum of a never-ending series of terms.

3. What was aleph naught?
(a) A symbol to represent the number of items in a set.
(b) A method to determine the sum of a series.
(c) A method to numerate terms.
(d) A symbol to state the sum of a series.

4. What was most noticeable about Euler at a young age?
(a) He had a remarkable memory.
(b) He was very athletic.
(c) He had an aptitude for literature.
(d) He was not very quick with arithmatic.

5. What series was Euler most famous for?
(a) 1 + 1/4 + 1/9 + 1/16 . . . + 1/k² . . .
(b) 1 + 1/2 + 3/4 + 4/5 . . .
(c) 1 + 1/2 + 1/6 + 1/10 + 1/15 . . .
(d) 1 + 1/2³ + 1/3³ + 1/4³ . . . 1/k³ . . .

Short Answer Questions

1. Where does the center of mathematical thinking shift to after Italy?

2. Where was Euler born?

3. How did Gauss feel about his best work?

4. What is one proof that Euler was able to prove?

5. In what area was Gauss especially interested?

Short Essay Questions

1. Explain what was the definition of a series before the Bernoullis, and give examples of what was known.

2. Explain how Gottfried Leibniz was able to publish his method of calculus.

3. Describe who were Jakob and Johann Bernoulli.

4. Describe some of Gauss's work.

5. What was Euler able to prove about 2²ⁿ + 1? Why was this a great accomplishment?

6. Who was Georg Cantor, and what was significant about his work in mathematics?

7. Explain why Eulers sum of π²/6 was in some ways surprising.

8. What great theorems and work of Newton did Dunham highlight?

9. Explain any methods used by Cantor that were unsuccessful.

10. Describe the great theorem explained by Dunham in this chapter.

(see the answer keys)

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