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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. Which of the following is a series that the Bernoullis proposed did not converge on a finite sum?**
**(a)** 1 + 2 + 3 + 4 + 5. . . **(b)** 1 + 1/2 + 1/6 + 1/10 + 1/15 . . . **(c)** 1 + 1/2 + 3/4 + 4/5 . . . **(d)** 1 + 1/2 + 1/3 + 1/4 + 1/5 . . . 1/1000 . . .

**2. What did Euler prove about 2²ⁿ + 1?**
**(a)** That the statement is always a prime number. **(b)** That the statment is sometimes prime and sometimes composite. **(c)** That the statement is neither prime nor composite. **(d)** That the statement is always a composite number.

**3. What was true when Euler used n = 5 in the statement 2²ⁿ + 1?**
**(a)** The statement was a composite number. **(b)** The statement was a prime number. **(c)** The statement was not a prime number. **(d)** The statement was a perfect number.

**4. Where did Euler study at the age of 20?**
**(a)** University of Moscow. **(b)** Oxford. **(c)** Cambrigde. **(d)** The Academy in St. Petersburg.

**5. What did George Cantor determine to be true of a set of rational numbers?**
**(a)** They are all prime numbers. **(b)** They are denumerable. **(c)** They are all composite numbers. **(d)** They are non-denumerable.

## Short Answer Questions

**1.** Who else, besides Newton, independently discovered a calculus method?

**2.** What great theorem is presented by Dunham in this chapter?

**3.** What did Euler's sum surprisingly connect?

**4.** What did Cantor define as the continuum?

**5.** What is the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ ... 1/k³ . . .?

## Short Essay Questions

**1.** Describe the connection between Fermat and Euler's work.

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

**3.** What was Gauss's major unpublished achievement in geometry?

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

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

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

**7.** Describe the great theorem explained by Dunham in this chapter.

**8.** Describe Newton's days in Cambridge and what he eventually came to discover.

**9.** Describe Cantor's difficult personal life.

**10.** Describe what mathematical and artistic movements are focused on in the second half of the 19th century.

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