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

Multiple Choice Questions

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

2. On who's work did Euler base his number theory?
(a) Bernoulli's.
(b) Leibniz's.
(c) Fermat's.
(d) Newton's.

3. How did Euler prove if the number 4,294,967,297 was prime or composite?
(a) He used his own rule of squares.
(b) He used Newton's calulus methods.
(c) He divided it by 2.
(d) He factored it.

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

5. Where did George Cantor live in the 1860s and 1870s?
(a) Germany.
(b) Scotland.
(c) Britian.
(d) Russia.

6. Where did Newton go to school before he went to Cambridge?
(a) The King's School.
(b) Oxford Grammar School.
(c) Charles II Grammar School.
(d) Cambridge Prep.

7. What sum did Euler find for the series?
(a) π²/6
(b) The sum was infinite.
(c) 1.
(d) 2.

8. 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 squared numbers.
(c) It's a convergent series of cubic numbers.
(d) It's a divergent series with a sum of 2.

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

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

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

12. What did Dunham describe as lacking from calculus previous to the mid-19th century?
(a) Definitions of infinately large and small quantities.
(b) An explanation of non-Eulidean mathematics.
(c) Foundations that link it to the principles of geometry.
(d) Description of the word "area."

13. 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 not denumerable.
(d) They are denumerable,

14. What did most of 19th century mathematics focus on, as highlighted by Dunham?
(a) The immediately practical.
(b) The theoretical.
(c) Geometry.
(d) Algebra.

15. What is true about the successive squared denominator series proposed by the Bernoullis?
(a) The sum diverges.
(b) The sum converges to 2.
(c) The sum converges.
(d) The sum diverges into infinity.

Short Answer Questions

1. What hindered Euler's work as he grew older?

2. Where did Euler study at the age of 20?

3. What did Cantor's work do to mathematics?

4. What did Cantor develop?

5. What did Newton's calculus involve?

(see the answer keys)

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