Bruce H Edwards
Author
Series
Language
English
Description
Does the celebrated harmonic series diverge or converge? Discover a proof using the integral test. Then generalize to define an entire class of series called p-series, and prove a theorem showing when they converge. Close with the sum of the harmonic series, the fascinating Euler-Mascheroni constant, which is not known to be rational or irrational.
Author
Series
Language
English
Description
Put your precalculus skills to use by splitting up complicated algebraic expressions to make them easier to integrate. Learn how to deal with linear factors, repeated linear factors, and irreducible quadratic factors. Finally, apply these techniques to the solution of the logistic differential equation.
Author
Series
Great Courses volume 2
Language
English
Description
The model for modern mathematical thinking was forged 2,300 years ago in Euclid's Elements. Prove three of Euclid's theorems and investigate his famous fifth postulate dealing with parallel lines. Also, learn how proofs are important in Professor Edwards's own research.
Author
Series
Great Courses volume 12
Language
English
Description
What does the decimal 0.99999... forever equal? Is it less than 1? Or does it equal 1? Apply mathematical induction to geometric series to find the solution. Also use induction to find the formulas for other series, including factorials, which are denoted by an integer followed by the "!" sign.
Author
Series
Great Courses volume 4
Language
English
Description
Continue your study of logic by looking at negations of statements and the logical operation called implication, which is used in most mathematical theorems. Professor Edwards opens the lecture with a fascinating example of the implication of a false hypothesis that appears to pose a logical puzzle.
Author
Series
Great Courses volume 9
Language
English
Description
Explore elementary set theory, learning the concepts and notation that allow manipulation of sets, their unions, their intersections, and their complements. Then try your hand at proving that two sets are equal, which involves showing that each is a subset of the other.
Author
Series
Great Courses volume 7
Language
English
Description
Probe the power of one of the most popular techniques for proving theorems - proof by contradiction. Begin by constructing a truth table for the contrapositive. Then work up to Euclid's famous proof that answers the question: Can the square root of 2 be expressed as a fraction?
Author
Series
Great Courses volume 20
Language
English
Description
The great mathematician Carl Friedrich Gauss once said that if mathematics is the queen of the sciences, then number theory is the queen of mathematics. Embark on the study of this fascinating discipline by proving theorems about prime numbers.
Author
Series
Great Courses volume 8
Language
English
Description
Start with the simple case of an isosceles triangle, defined as having two equal sides or two equal angles. Discover that equal sides and equal angles apply to all isosceles triangles and are an example of an "if-and-only-if" theorem, which occurs throughout mathematics.
Author
Series
Great Courses volume 22
Language
English
Description
Use different proof techniques to explore square and triangular numbers. Square numbers are numbers such as 1, 4, 9, and 16 that are the squares of integers. Triangular numbers represent the total dots needed to form an equilateral triangle, such as 1, 3, 6, and 10.
Author
Series
Great Courses volume 3
Language
English
Description
Logic is the foundation of mathematical proofs. In the first of three lectures on logic, study the connectors "and" and "or." When used in combination in mathematical statements, these simple terms can create interesting complexity. See how truth tables are very useful for determining when such statements are true or false.
Author
Series
Great Courses volume 5
Language
English
Description
In the final lecture on logic, explore the quantifiers "for all" and "there exists," learning how these operations are negated. Quantifiers play a large role in calculus - for example, when defining the concept of a sequence, which you study in greater detail in upcoming lectures.
Author
Series
Great Courses volume 18
Language
English
Description
Strengthen your appreciation for good proofs by looking at bad proofs, including common errors that students make such as dividing by 0 and circular reasoning. Then survey the history of attempts to prove some renowned conjectures from geometry and number theory.
Author
Series
Great Courses volume 23
Language
English
Description
Investigate the intriguing link between perfect numbers and Mersenne primes. A number is perfect if it equals the sum of all its divisors, excluding itself. Mersenne primes are prime numbers that are one less than a power of 2. Oddly, the known examples of both classes of numbers are 47.
Author
Series
Great Courses volume 24
Language
English
Description
Prove some properties of the celebrated number e, the base of the natural logarithm, which plays a crucial role in precalculus and calculus. Close with a challenging proof testing whether e is rational or irrational - just as you did with the square root of 2 in Lecture 7.
Author
Series
Great Courses volume 21
Language
English
Description
Dig deeper into prime numbers and number theory by proving a conjecture that asserts that there are arbitrarily large gaps between successive prime numbers. Then turn to some celebrated conjectures in number theory, which are easy to state but which have withstood all attempts to prove them.
Author
Series
Great Courses volume 14
Language
English
Description
Analyze existence proofs, which show that a mathematical object must exist, even if the actual object remains unknown. Close with an elegant and subtle argument proving that there exists an irrational number raised to an irrational power, and the result is a rational number.
Author
Series
Great Courses volume 11
Language
English
Description
In the first of three lectures on mathematical induction, try out this powerful tool for proving theorems about the positive integers. See how an inductive proof is like knocking over a row of dominos: Once the base case pushes over a second case, then by induction all cases fall.
Author
Series
Great Courses volume 10
Language
English
Description
Tackle infinite sets, which pose fascinating paradoxes. For example, the set of integers is a subset of the set of rational numbers, and yet there is a one-to-one correspondence between them. Explore other properties of infinite sets, proving that the real numbers between 0 and 1 are uncountable.
Author
Series
Great Courses volume 17
Language
English
Description
You've studied proofs. How about disproofs? How do you show that a conjecture is false? Experience the fun of finding counterexamples. Then explore some famous paradoxes in mathematics, including Bertrand Russell's barber paradox, which shook the foundations of set theory.