|Name: _________________________||Period: ___________________|
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?
This section contains 789 words
(approx. 3 pages at 300 words per page)