Bruce H Edwards
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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.
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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.
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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.
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Great Courses volume 16
Language
English
Description
The famous Four Color theorem, dealing with the minimum number of colors needed to distinguish adjacent regions on a map with different colors, was finally proved by a brute force technique called enumeration of cases. Learn how this approach works and why mathematicians dislike it - although they often rely on it.
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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.
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Great Courses volume 6
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English
Description
Begin a series of lectures on different proof techniques by looking at direct proofs, which make straightforward use of a hypothesis to arrive at a conclusion. Try several examples, including proofs involving division and inequalities. Then learn tricks that mathematicians use to make proofs easier than they look.
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Great Courses volume 17
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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.
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Great Courses volume 11
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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.
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Great Courses volume 1
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English
Description
Start by proving that two odd numbers multiplied together always give an odd number. Next, look ahead at some of the intriguing proofs you will encounter in the course. Then explore the characteristics of a proof and tips for improving your skill at proving mathematical theorems.
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Great Courses volume 19
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English
Description
Before he became the 20th U. S. president, James A. Garfield published an original proof of the Pythagorean theorem that relied on a visual argument. See how pictures play an important role in understanding why a particular mathematical statement may be true. But is a visual proof really a proof?
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Great Courses volume 24
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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.
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Great Courses volume 23
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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.
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Great Courses volume 21
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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.
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Great Courses volume 15
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English
Description
How do you prove that a given mathematical result is unique? Assume that more than one solution exists and then see if there is a contradiction. Use this technique to prove several theorems, including the important division algorithm from arithmetic.
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Great Courses volume 13
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English
Description
Use a technique called strong induction to prove an elementary theorem about prime numbers. Next, apply strong induction to the famous Fibonacci sequence, verifying the Binet formula, which can specify any number in the sequence. Test the formula by finding the 21-digit-long 100th Fibonacci number.
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Great Courses volume 12
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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.
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Great Courses volume 9
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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.
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Great Courses volume 2
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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.
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Great Courses volume 20
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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.
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Great Courses volume 14
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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.