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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 infinite. **(b)** That the set is equal to 1. **(c)** That the set is still equal to c. **(d)** That the series is infinite.

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

**3. Which of the following is not denumerable, proven by Cantor's theorem?**
**(a)** A set of a geometric series. **(b)** A set of transcendental numbers. **(c)** A set of rational numbers. **(d)** A set of imaginary numbers.

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

**5. What did Cantor define as the continuum?**
**(a)** All imaginary and real numbers. **(b)** All imaginary numbers. **(c)** The square root of any real number. **(d)** Real numbers between 0 and 1.

## Short Answer Questions

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

**2.** Who was Euler's teacher?

**3.** Where was Euler born?

**4.** When was Euler born?

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

## Short Essay Questions

**1.** Explain how Gottfried Leibniz was able to publish his method of calculus.

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

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

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

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

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

**7.** What was Gauss's major unpublished achievement in geometry?

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

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

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

This section contains 817 words(approx. 3 pages at 300 words per page) |