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

2. What did most of 19th century mathematics focus on, as highlighted by Dunham?
(a) The theoretical.
(b) Geometry.
(c) Algebra.
(d) The immediately practical.

3. What did Cantor suspect about transfinite cardinals?
(a) That there are transfinite cardinals much less than c.
(b) That the sum of the series is finite.
(c) That the sum of the series is infinite.
(d) That there are transfinite cardinals even greater than c.

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

5. Who was Euler's teacher?
(a) Isaac Newton.
(b) Gottfried Leibniz.
(c) Johann Bernoulli.
(d) Jakob Bernoulli.

Short Answer Questions

1. What did Cantor's beliefs lead him to think?

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

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

4. What hindered Euler's work as he grew older?

5. What didn't Euler attempt?

Short Essay Questions

1. Give an example of a series who's sum is still unknown.

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

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

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

5. Summarize in a few sentences, what types of number sets did Cantor prove to be denumerable and non-denumerable.

6. Describe Cantor's difficult personal life.

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

8. Why did Euler start working on the sum of series?

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

10. What were the two transfinite cardinals discovered by Cantor, and what method did he use to determine them?

(see the answer keys)

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