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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 is the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ ... 1/k³ . . .?
(a) Nobody has determined the sum.
2. 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 . . .
3. What did Cantor define as the continuum?
(a) The square root of any real number.
(b) All imaginary and real numbers.
(c) All imaginary numbers.
(d) Real numbers between 0 and 1.
4. What was the same about the series proposed by Leibniz and the series proposed by Bernoulli?
(a) Both series were convergent.
(b) Both series were composed of successively larger terms.
(c) Both series were divergent.
(d) Both series were composed of successively smaller terms.
5. What did Cantor's work do to mathematics?
(a) It caused much agreement among mathematicians on the use of calculus.
(b) It raised arguments on the origins of geometry.
(c) It forced the reexamination of set theory.
(d) It caused a reevaluation of basic algebra.
Short Answer Questions
1. Who was Euler's teacher?
2. To how many decimal places did Newton determine the number for pi?
3. What did Cantor find after extending the continuum between 0 and 1 into two dimensions?
4. What did George Cantor discover?
5. In what area was Gauss especially interested?
Short Essay Questions
1. What great theorems and work of Newton did Dunham highlight?
2. Describe the controversy that Newton was caught in with his publication of his calculus methods.
3. What was the great theorem of this chapter? Describe it briefly.
4. Summarize in a few sentences, what types of number sets did Cantor prove to be denumerable and non-denumerable.
5. Explain why Eulers sum of Ï€Â²/6 was in some ways surprising.
6. Describe the great theorem explained by Dunham in this chapter.
7. What was Gauss's major unpublished achievement in geometry?
8. What were the two transfinite cardinals discovered by Cantor, and what method did he use to determine them?
9. Explain what was the definition of a series before the Bernoullis, and give examples of what was known.
10. Describe Newton's days in Cambridge and what he eventually came to discover.
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