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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. 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 sum of a never-ending series of terms.
(d) The finite sum of a divergent series.

2. What did Cantor's cardinal numbers represent?
(a) Sets of all imaginary numbers.
(b) Finite series.
(c) Infinite sets.
(d) Series of prime numbers.

3. When was Euler born?
(a) 1903.
(b) 1707.
(c) 1796.
(d) 1658.

4. What was Dunham central theorem for this chapter?
(a) That the area of a circle is fundamentally related to the square of its area.
(b) That the sum of an infintire series is always infinite.
(c) That there are other transfinite cardinals greater than c.
(d) That the sum of a set of real numbers is finite.

5. In what area was Gauss especially interested?
(a) The proof of the infinite series.
(b) The elements of geometry.
(c) The elements of number theory.
(d) The circumference of Earth.

Short Answer Questions

1. What was aleph naught?

2. What did Dunham describe as lacking from calculus previous to the mid-19th century?

3. What was true when Euler used n = 5 in the statement 2²ⁿ + 1?

4. What did Cantor define as the continuum?

5. Where did Newton go to school before he went to Cambridge?

Short Essay Questions

1. Where did the center of mathematical thinking shift to in the !7th Century, and who are the major scholars of this time period?

2. Describe some of Gauss's work.

3. Describe Cantor's difficult personal life.

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

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

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

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

8. Describe the connection between Fermat and Euler's work.

9. Describe some of the characteristics of Leonhard Euler, and what made him successful.

10. What was the great theorem of this chapter? Describe it briefly.

(see the answer keys)

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