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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. **(b)** 2. **(c)** 0. **(d)** Infinity.

**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**7.** What was Gauss's major unpublished achievement in geometry?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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