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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 were the main technique(s) that Euler used to find the sum of the series?**
**(a)** Calculus methods. **(b)** Cubic equations. **(c)** Quadratic sums, **(d)** Trigonometry and basic algebra.

**2. Which of the following was one of Gauss' early discoveries?**
**(a)** A method to simplify Newton's calulus. **(b)** A demonstration of the sum of the series 1 + 1/2³ + 1/3³ + 1/4³ . . . 1/k³ . . **(c)** A way to construct a regular 17-sided polygon. **(d)** A proof that the Pythagorean Theorem was correct.

**3. Who else, besides Newton, independently discovered a calculus method?**
**(a)** Gottfried Leibniz. **(b)** Isaac Barrow. **(c)** Pierre de Fermat. **(d)** John Napier.

**4. What series was Euler most famous for?**
**(a)** 1 + 1/2 + 1/6 + 1/10 + 1/15 . . . **(b)** 1 + 1/4 + 1/9 + 1/16 . . . + 1/k² . . . **(c)** 1 + 1/2³ + 1/3³ + 1/4³ . . . 1/k³ . . . **(d)** 1 + 1/2 + 3/4 + 4/5 . . .

**5. Which of the following demonstrates the successive squared denominator series?**
**(a)** 1 + 1/4 + 1/9 + 1/16 . . . **(b)** 1 + 1/2 + 3/4 + 4/5 . . . **(c)** 1 + 1/2 + 1/6 + 1/10 + 1/15 . . . **(d)** 1 + 1/2 + 1/3 + 1/4 + 1/5 . . . 1/1000 . . .

## Short Answer Questions

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

**2.** What is true about real numbers between 0 and 1?

**3.** What did Newton's calculus involve?

**4.** What did Cantor develop?

**5.** Which phrase best describes Newton as a student at Cambridge?

## Short Essay Questions

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

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

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

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

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

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

**7.** Explain why Eulers sum of Ï€Â²/6 was in some ways surprising.

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

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

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

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