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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 did Cantor find after extending the continuum between 0 and 1 into two dimensions?**
**(a)** That the set is equal to 1. **(b)** That the set is still equal to c. **(c)** That the set is infinite. **(d)** That the series is infinite.

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

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

**4. Who were Johann and Jakob Bernoulli?**
**(a)** Twin brothers and students of Newton. **(b)** Brothers and students of Leibniz. **(c)** Cousins and students with Leibniz in Paris. **(d)** Cousins and students with Newton at Cambridge.

**5. What was described as true about the series 1 + 1/2 + 1/6 + 1/10 + 1/15 + 1/21?**
**(a)** It's a convergent series of cubic numbers. **(b)** It's a divergent series with a sum of 2. **(c)** It's a convergent series of triangular numbers. **(d)** It's a divergent series squared numbers.

## Short Answer Questions

**1.** What is true about the successive squared denominator series proposed by the Bernoullis?

**2.** Which of the following is a series that the Bernoullis proposed did not converge on a finite sum?

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

**4.** What was Dunham central theorem for this chapter?

**5.** What was most noticeable about Euler at a young age?

## Short Essay Questions

**1.** Describe who were Jakob and Johann Bernoulli.

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

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

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

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

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

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

**8.** Explain any methods used by Cantor that were unsuccessful.

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

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

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