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This test consists of 5 short answer questions, 10 short essay questions, and 1 (of 3) essay topics.

## Short Answer Questions

**1.** Where did George Cantor live in the 1860s and 1870s?

**2.** Which word best describes Newton's childhood?

**3.** Which of the following was a major part of Gauss' work in mathematics?

**4.** What did Euler prove about 2²ⁿ + 1?

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

## Short Essay Questions

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

**2.** Describe what the Bernoullis discovered about series, and give an example.

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

**4.** Describe some of Gauss's work.

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

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

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

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

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

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

## Essay Topics

Write an essay for ONE of the following topics:

### Essay Topic 1

Summarize the discoveries on series made by the Bernoulli brothers and later, Euler. How did the Bernoullis advance mathematical understanding of an infinite series, and how did Euler even further advance this knowledge? Give examples. What principles about series and the sum of series are still being worked out in modern mathematics?

### Essay Topic 2

Write an essay to describe what was known about mathematics and geometry before and in the time of Thales.

Part 1) Describe what we know about geometry before Thales.

Part 2) In what ways were mathematics and geometry used before Thales?

Part 3) Explain what we know about Thales' work. How did it advance mathematics?

### Essay Topic 3

Summarize the puzzles that were presented as a result of Euclid's infinitude of primes. Name and describe some of these puzzles from a historical perspective, then explain what is known about each in modern mathematics. What puzzles remain unsolved today?

This section contains 821 words(approx. 3 pages at 300 words per page) |