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This test consists of 15 multiple choice questions and 5 short answer questions.

## Multiple Choice Questions

**1. Where was Euler born?**
**(a)** Finland. **(b)** Switzerland. **(c)** Denmark. **(d)** Germany.

**2. What did Newton's calculus involve?**
**(a)** Determining the area under a curve. **(b)** Proving the existance of pi. **(c)** Determining the volume of a sphere. **(d)** Proving the cubic equation.

**3. 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 triangular numbers. **(b)** It's a divergent series with a sum of 2. **(c)** It's a divergent series squared numbers. **(d)** It's a convergent series of cubic numbers.

**4. What great theorem is presented by Dunham in this chapter?**
**(a)** A theorem on finite series developed by Johann Bernoulli. **(b)** A theorem on series developed by Jakob and published by Johann Bernoulli. **(c)** A theorem on infinite series published by Jakob Bernoulli. **(d)** An improvement on Leibniz's caluclus as presented by Jakob Bernoulli.

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

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

**7. What is true about real numbers between 0 and 1?**
**(a)** No sum can be determined. **(b)** There is no set for these numbers. **(c)** They are denumerable, **(d)** They are not denumerable.

**8. Who, in modern day, is given credit for the calculus method?**
**(a)** Newton, **(b)** Johann Bernoulli. **(c)** Leibniz. **(d)** Both Newton and Leibniz.

**9. What did Cantor's cardinal numbers represent?**
**(a)** Infinite sets. **(b)** Finite series. **(c)** Series of prime numbers. **(d)** Sets of all imaginary numbers.

**10. Who encourages Newton during his studies at Cambridge?**
**(a)** Isaac Barrow. **(b)** Henry Stokes. **(c)** Henry Briggs. **(d)** John Napier.

**11. What did George Cantor discover?**
**(a)** A way to determine the accuracy of a calculation. **(b)** A way to compare the relative sizes of infinite sets. **(c)** A method to measure infinity. **(d)** A method to measure a curved area.

**12. How did Cantor finally prove his theory?**
**(a)** By extension of the infinite series. **(b)** By extension of the Pythagorean Theorem. **(c)** By using basic algebra. **(d)** By refining and expanding set theory.

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

**14. What was the same about the series proposed by Leibniz and the series proposed by Bernoulli?**
**(a)** Both series were composed of successively larger terms. **(b)** Both series were convergent. **(c)** Both series were composed of successively smaller terms. **(d)** Both series were divergent.

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

## Short Answer Questions

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

**2.** What did mathematicians want to perfect in the mid-19th century?

**3.** What did George Cantor determine to be true of a set of rational numbers?

**4.** What is one proof that Euler was able to prove?

**5.** What did Cantor's work do to mathematics?

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