THE PARTIAL SUMS OF THE HARMONIC SERIES

... we conclude that every δn is a positive number not exceeding 1. Observe that δn − δn+1 = (Hn − ln n) − (Hn+1 − ln(n + 1)) ...

... we conclude that every δn is a positive number not exceeding 1. Observe that δn − δn+1 = (Hn − ln n) − (Hn+1 − ln(n + 1)) ...

Math 142 Group Projects

... For this project you must work in groups of at most 3 students. You and your group will have to pick a topic by Friday, April 15. No two groups may present the same topic and the topics are awarded on a first come first serve basis. Each group will have 10-12 minutes to present their topic on Friday ...

... For this project you must work in groups of at most 3 students. You and your group will have to pick a topic by Friday, April 15. No two groups may present the same topic and the topics are awarded on a first come first serve basis. Each group will have 10-12 minutes to present their topic on Friday ...

Euler and the Exponential Base e

... Euler and the Exponential Base e In the next generation after Newton, Euler made extensive use of Newton's generalized binomial expansions greatly extending their range and utility. Newton used tables to construct infinite series, but once the method of formation of this series had been made clear E ...

... Euler and the Exponential Base e In the next generation after Newton, Euler made extensive use of Newton's generalized binomial expansions greatly extending their range and utility. Newton used tables to construct infinite series, but once the method of formation of this series had been made clear E ...

4 - Mathematics Department People Pages

... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. The first pe ...

... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, or σ(n) = 2 n. The first pe ...

Solovay-Strassen Primality Test If n is an odd natural number, then n

... due: 4pm Tuesday, May 23 ...

... due: 4pm Tuesday, May 23 ...

PDF

... which is one more than 561 times 34628679005806222298029624007549356624641833138184305191654410155773 Hence 561 is an Euler pseudoprime. The next few Euler pseudoprimes to base 2 are 1105, 1729, 1905, 2047, 2465, 4033, 4681 (see A047713 in Sloane’s OEIS). An Euler pseudoprime is sometimes called an ...

... which is one more than 561 times 34628679005806222298029624007549356624641833138184305191654410155773 Hence 561 is an Euler pseudoprime. The next few Euler pseudoprimes to base 2 are 1105, 1729, 1905, 2047, 2465, 4033, 4681 (see A047713 in Sloane’s OEIS). An Euler pseudoprime is sometimes called an ...

Ordinary Diferential Equations Computational Physics Ordinary Diferential Equations

... comparing solution to the true answer we see that there are small errors! Worse yet, the errors get BIGGER when we take smaller steps. What's going on?? ...

... comparing solution to the true answer we see that there are small errors! Worse yet, the errors get BIGGER when we take smaller steps. What's going on?? ...

Phasors

... This is just a simple phase shift, say, = t, = 0. We have Euler's identity: that relates trigonometric functions to complex exponentials, the math of which is a bit simpler: ...

... This is just a simple phase shift, say, = t, = 0. We have Euler's identity: that relates trigonometric functions to complex exponentials, the math of which is a bit simpler: ...

1 A Brief History of √−1 and Complex Analysis

... (1526–1572) wrote of a “wild idea” in his Algebra of 1542: Formal manipu√ lation of −1 to obtain real results. • François Viète (1540–1603) used cosine and arccosine to avoid imaginary quantities. • Leonhard Euler (1707–1783) wrote to Johann Bernoulli (his former teacher) in 1740 the result now kn ...

... (1526–1572) wrote of a “wild idea” in his Algebra of 1542: Formal manipu√ lation of −1 to obtain real results. • François Viète (1540–1603) used cosine and arccosine to avoid imaginary quantities. • Leonhard Euler (1707–1783) wrote to Johann Bernoulli (his former teacher) in 1740 the result now kn ...

MATHEMATICS ENRICHMENT CLUB.1 Problem Sheet 16

... 2. Show that any straight line passing through the centre of a parallelogram (i.e. the intersection of the diagonals) divides the parallelogram into two equal areas. 3. (a) Show that any convex polygon of area 1 can be enclosed in a parallelogram of area 2. (b) Show that a triangle of area 1 cannot ...

... 2. Show that any straight line passing through the centre of a parallelogram (i.e. the intersection of the diagonals) divides the parallelogram into two equal areas. 3. (a) Show that any convex polygon of area 1 can be enclosed in a parallelogram of area 2. (b) Show that a triangle of area 1 cannot ...

Leonhard Euler - UT Mathematics

... Father: Paul Euler, Calvinist pastor Mother: Marguerite Brucker, daughter of a pastor • Married-Twice: 1)Katharina Gsell, 2)her half sister • Children-Thirteen (three outlived him) ...

... Father: Paul Euler, Calvinist pastor Mother: Marguerite Brucker, daughter of a pastor • Married-Twice: 1)Katharina Gsell, 2)her half sister • Children-Thirteen (three outlived him) ...

Full text

... a one-parameter family of identities that are valid for all n > 0. B. REMARKS By using the first equation given in (1), other identities, involving Fibonacci numbers and Euler numbers, can be found if &2k in terms of Euler numbers is inserted in the identities obtained in [ 1 ] . Since ...

... a one-parameter family of identities that are valid for all n > 0. B. REMARKS By using the first equation given in (1), other identities, involving Fibonacci numbers and Euler numbers, can be found if &2k in terms of Euler numbers is inserted in the identities obtained in [ 1 ] . Since ...

Slides

... It may have the appearance, after the fact, of being a perfectly natural “analytic continuation,” so to speak, of a concept—such as the development of zero and negative numbers as an expansion of whole numbers, and from there: rational numbers, etc. ...

... It may have the appearance, after the fact, of being a perfectly natural “analytic continuation,” so to speak, of a concept—such as the development of zero and negative numbers as an expansion of whole numbers, and from there: rational numbers, etc. ...

March - The Euler Archive - Mathematical Association of America

... Euclid wondered this more than 200 years ago, and his proof that “Prime numbers are more than any assigned multitude of prime numbers” (Elements IX.20) is often considered one of the most beautiful proofs in all of mathematics.1 Prime numbers, their detection, their frequency and their special prope ...

... Euclid wondered this more than 200 years ago, and his proof that “Prime numbers are more than any assigned multitude of prime numbers” (Elements IX.20) is often considered one of the most beautiful proofs in all of mathematics.1 Prime numbers, their detection, their frequency and their special prope ...

Approximation of Euler Number Using Gamma Function

... The number e is a mathematical constant and its value is approximately equal to 2.71828, which is popularly known as Euler number or Napier constant. Its symbol is given in the honour of famous Swiss mathematician Leonhard Euler. The constant is not be confused with Euler-Mascheroni constant or simp ...

... The number e is a mathematical constant and its value is approximately equal to 2.71828, which is popularly known as Euler number or Napier constant. Its symbol is given in the honour of famous Swiss mathematician Leonhard Euler. The constant is not be confused with Euler-Mascheroni constant or simp ...

q - Personal.psu.edu - Penn State University

... Historical Motivation In 1740, Philippe Naude sent a letter to Leonhard Euler asking the following: “How many ways can the number 50 be written as a sum of seven different positive integers?” ...

... Historical Motivation In 1740, Philippe Naude sent a letter to Leonhard Euler asking the following: “How many ways can the number 50 be written as a sum of seven different positive integers?” ...

Set 01 - Joseph Wells

... d. a, b, c, d, e, f, g, h, i, j, k, l, m, n, o | abc, bde, df g, f hi, haj, c−1 kl, e−1 mn, g −1 ok −1 , i−1 l−1 m−1 , j −1 n−1 o−1 2. Prove that there are only five regular polyhedra (the regular tetrahedron, cube, octahedron, dodecahedron, icosahedron) by subdividing the sphere into n-gons such th ...

... d. a, b, c, d, e, f, g, h, i, j, k, l, m, n, o | abc, bde, df g, f hi, haj, c−1 kl, e−1 mn, g −1 ok −1 , i−1 l−1 m−1 , j −1 n−1 o−1 2. Prove that there are only five regular polyhedra (the regular tetrahedron, cube, octahedron, dodecahedron, icosahedron) by subdividing the sphere into n-gons such th ...

Euler`s Identity

... different areas of work and study. Leonhard Euler, born in Basel, Switzerland, in 1707, was fortunate to begin his life with a well-rounded education, considering his father’s wish that Euler take up ministry as his livelihood. Euler had the good fortune in his childhood to come into contact with pe ...

... different areas of work and study. Leonhard Euler, born in Basel, Switzerland, in 1707, was fortunate to begin his life with a well-rounded education, considering his father’s wish that Euler take up ministry as his livelihood. Euler had the good fortune in his childhood to come into contact with pe ...

Transcendental vs. Algebraic Numbers

... History of the Number e • It first appeared in 1618 in an appendix probably by Oughred, it was a table giving natural logs (1, pg 1) • In 1624, Briggs gave a numerical approximation to the base 10 log of e but didn’t mention e itself (1, pg 1) • In 1647 Saint Vincent computed the area under a rectan ...

... History of the Number e • It first appeared in 1618 in an appendix probably by Oughred, it was a table giving natural logs (1, pg 1) • In 1624, Briggs gave a numerical approximation to the base 10 log of e but didn’t mention e itself (1, pg 1) • In 1647 Saint Vincent computed the area under a rectan ...

Three

... digits. If you didn’t know anything about continued√fractions, you would think that 12599/10000 is a good approximation to 3 2. Find its cost. See if you can find any rational number that is not a convergent but still has low cost. (4) Euler’s rule. Let us use the notation ha0 , a1 , . . . , ak i to ...

... digits. If you didn’t know anything about continued√fractions, you would think that 12599/10000 is a good approximation to 3 2. Find its cost. See if you can find any rational number that is not a convergent but still has low cost. (4) Euler’s rule. Let us use the notation ha0 , a1 , . . . , ak i to ...

Mathematical Diversions

... One thing one can observe is that the card that was in position k, after the shuffle ends in position 2k mod 9. This suggests or shows that the number of shuffles that take the deck back to its original position is the first n such that 2n = 1 mod 9. That number turns out to be 6. One might remember ...

... One thing one can observe is that the card that was in position k, after the shuffle ends in position 2k mod 9. This suggests or shows that the number of shuffles that take the deck back to its original position is the first n such that 2n = 1 mod 9. That number turns out to be 6. One might remember ...

Leonhard Euler (1707 – 1783) (/ˈɔɪlər/ OY-lər; German pronunciation: [ˈɔʏlɐ], local pronunciation: [ˈɔɪlr̩]) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.Euler is considered to be the preeminent mathematician of the 18th century and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians; his collected works fill 60 to 80 quarto volumes. He spent most of his adult life in St. Petersburg, Russia, and in Berlin, then the capital of Prussia.A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: ""Read Euler, read Euler, he is the master of us all.""