Sum of Generalized Alternating Harmonic Series with Three Periodically Repeated Numerators
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Mathematical Constants and Sequences
Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) .. -
Survey of Math: Chapter 21: Consumer Finance Savings Page 1
Survey of Math: Chapter 21: Consumer Finance Savings Page 1 Geometric Series Consider the following series: 1 + x + x2 + x3 + ··· + xn−1. We need to figure out a way to write this without the ···. Here’s how: s = 1 + x + x2 + x3 + ··· + xn−1 subtract xs = x + x2 + x3 + ··· + xn−1 + xn s − sx = 1 − 0 − 0 − 0 − · · · − 0 − xn s − sx = 1 − xn Now solve for s: 1 − xn xn − 1 s(1 − x) = 1 − xn ⇒ s = = . 1 − x x − 1 xn − 1 Therefore, a geometric series has the following sum: 1 + x + x2 + x3 + ··· + xn−1 = . x − 1 Exponential and Natural Logarithms As we have seen, some of our equations involve exponents. To effectively deal with exponents, we need to be able to work with exponentials and logarithms. The exponential and natural logarithm functions are inverse functions, and related by the following rules 1 m e = 1 + for m large m e = 2.71828 ... ln(eA) = A where A is a constant eln(A) = A If the base of the exponent is not e, we can use the following rule: ln(bA) = A ln(b) where A and b are constants ln(2A) = A ln(2) ln((1 + r)A) = A ln(1 + r) Note: • The natural logarithm acts on a number, so we read ln(2) as “The natural logarithm of 2”. • We would never write ln(2) = 2 ln, since this loses the fact that the natural logarithm must act on something. It is a functional evaluation, not a multiplication. • This is similar to trigonometric functions, like sin(π). -
MATHEMATICAL INDUCTION SEQUENCES and SERIES
MISS MATHEMATICAL INDUCTION SEQUENCES and SERIES John J O'Connor 2009/10 Contents This booklet contains eleven lectures on the topics: Mathematical Induction 2 Sequences 9 Series 13 Power Series 22 Taylor Series 24 Summary 29 Mathematician's pictures 30 Exercises on these topics are on the following pages: Mathematical Induction 8 Sequences 13 Series 21 Power Series 24 Taylor Series 28 Solutions to the exercises in this booklet are available at the Web-site: www-history.mcs.st-andrews.ac.uk/~john/MISS_solns/ 1 Mathematical induction This is a method of "pulling oneself up by one's bootstraps" and is regarded with suspicion by non-mathematicians. Example Suppose we want to sum an Arithmetic Progression: 1+ 2 + 3 +...+ n = 1 n(n +1). 2 Engineers' induction € Check it for (say) the first few values and then for one larger value — if it works for those it's bound to be OK. Mathematicians are scornful of an argument like this — though notice that if it fails for some value there is no point in going any further. Doing it more carefully: We define a sequence of "propositions" P(1), P(2), ... where P(n) is "1+ 2 + 3 +...+ n = 1 n(n +1)" 2 First we'll prove P(1); this is called "anchoring the induction". Then we will prove that if P(k) is true for some value of k, then so is P(k + 1) ; this is€ c alled "the inductive step". Proof of the method If P(1) is OK, then we can use this to deduce that P(2) is true and then use this to show that P(3) is true and so on. -
Interesting Topics
Interesting Topics for Bachelor Theses Walter M. Bohm¨ Institute for Statistics and Mathematics Vienna University of Economics [email protected] October 4, 2018 The picture on the title page is an artwork by Jakob Bohm,¨ www.jacob-julian.com The other pictures used in this book are from Wikipedia Commons and MacTutor History of Mathematics archive maintained at The University of St. Andrews. 2 Foreword This booklet is a collection of topics which I prepared over the time for my students. The selection of topics reflects my personal interests and is therefore biased towards combinatorial mathematics, probability and statistics, opera- tions research, scheduling theory and, yes, history of mathematics. It is in the nature of things that the level of difficulty varies from topic to topic. Some are technically more demanding others somewhat easier. The only prerequisite to master these topics are courses in mathematics and statistics at an undergraduate university level. Otherwise, no special prior knowledge in mathematics is afforded. However, what is needed is serious interest in mathematics, of course. How is a Topic organized? Each topic consists of three parts: (1) An Invitation Of course, the major purpose of this invitation is to raise your interest and to draw your attention to a problem which I found very interesting, attractive and challenging. Further, in each invitation I introduce some basic terminology so that you can start reading basic literature related to the topic. (2) Where to go from here Some of my invitations are more detailed depending on the topic, so you may ask yourself: Is there anything left for me to do? Yes, there is lot of work still to be done. -
2. Infinite Series
2. INFINITE SERIES 2.1. A PRE-REQUISITE:SEQUENCES We concluded the last section by asking what we would get if we considered the “Taylor polynomial of degree for the function ex centered at 0”, x2 x3 1 x 2! 3! As we said at the time, we have a lot of groundwork to consider first, such as the funda- mental question of what it even means to add an infinite list of numbers together. As we will see in the next section, this is a delicate question. In order to put our explorations on solid ground, we begin by studying sequences. A sequence is just an ordered list of objects. Our sequences are (almost) always lists of real numbers, so another definition for us would be that a sequence is a real-valued function whose domain is the positive integers. The sequence whose nth term is an is denoted an , or if there might be confusion otherwise, an n 1, which indicates that the sequence starts when n 1 and continues forever. Sequences are specified in several different ways. Perhaps the simplest way is to spec- ify the first few terms, for example an 2, 4, 6, 8, 10, 12, 14,... , is a perfectly clear definition of the sequence of positive even integers. This method is slightly less clear when an 2, 3, 5, 7, 11, 13, 17,... , although with a bit of imagination, one can deduce that an is the nth prime number (for technical reasons, 1 is not considered to be a prime number). Of course, this method com- pletely breaks down when the sequence has no discernible pattern, such as an 0, 4, 3, 2, 11, 29, 54, 59, 35, 41, 46,.. -
Calculating Logs Manually
calculating logs manually File Name: calculating logs manually.pdf Size: 2113 KB Type: PDF, ePub, eBook Category: Book Uploaded: 3 May 2019, 12:21 PM Rating: 4.6/5 from 561 votes. Status: AVAILABLE Last checked: 10 Minutes ago! In order to read or download calculating logs manually ebook, you need to create a FREE account. Download Now! eBook includes PDF, ePub and Kindle version ✔ Register a free 1 month Trial Account. ✔ Download as many books as you like (Personal use) ✔ Cancel the membership at any time if not satisfied. ✔ Join Over 80000 Happy Readers Book Descriptions: We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer, you have convenient answers with calculating logs manually . To get started finding calculating logs manually , you are right to find our website which has a comprehensive collection of manuals listed. Our library is the biggest of these that have literally hundreds of thousands of different products represented. Home | Contact | DMCA Book Descriptions: calculating logs manually And let’s start with the logarithm of 2. As a kid I always wanted to know how to calculate log 2 and nobody was able to tell me. Can we guess the logarithm of 19,683.Let’s follow the chain. We are looking for 1225, so to account for the difference let’s round 3.0876 up to 3.088. How to calculate log 11. Here is a way to do this. We can take the geometric mean of 10 and 12. -
1 Introduction
1 Introduction Definition 1. A rational number is a number which can be expressed in the form a/b where a and b are integers with b > 0. Theorem 1. A real number α is a rational number if and only if it can be expressed as a repeating decimal, that is if and only if α = m.d1d2 . dkdk+1dk+2 . dk+r, where m = [α] if α ≥ 0 and m = −[|α|] if α < 0, where k and r are non-negative integers with r ≥ 1, and where the dj are digits. Proof. If α = m.d1d2 . dkdk+1dk+2 . dk+r, then (10k+r − 10k)α ∈ Z and it easily follows that α is rational. If α = a/b with a and b integers and b > 0, then α = m.d1d2 ... for some digits dj. If {x} denotes the fractional part of x, then j {10 |α|} = 0.dj+1dj+2 .... (1) On the other hand, {10j|α|} = {10ja/b} = u/b for some u ∈ {0, 1, . , b − 1}. Hence, by the pigeon-hole principle, there exist non-negative integers k and r with r ≥ 1 and {10k|α|} = {10k+r|α|}. From (1), we deduce that 0.dk+1dk+2 ··· = 0.dk+r+1dk+r+2 ... so that α = m.d1d2 . dkdk+1dk+2 . dk+r, and the result follows. Definition 2. A number is irrational if it is not rational. Theorem 2. A real number α which can be expressed as a non-repeating decimal is irrational. Proof 1. From the argument above, if α = m.d1d2 .. -
Introducing Mathematica
Introducing Mathematica John H. Lowenstein Preface This informal introduction to Mathematica®(a product of Wolfram Research, Inc.) is offered as a downloadable resource for users of the textbook Essentials of Hamiltonian Dynamics (Cambridge University Press, 2012). The aim is to familiarize the student with the core concepts and functions of Mathematica programming, so that he or she can very quickly become comfortable with computational methods in dealing with the illustrative examples and exercises in the textbook. The scope of Mathematica obviously greatly exceeds what can be covered in these few pages, and so it is highly recommended that the student take full advantage of the excellent documentation which is included with the software (accessible via the Help menu). 1. Getting started Mathematica conducts a dialogue with the user: you type in a mathematical expression, press Enter , and Mathemat- ica evaluates the expression according to rules which are either built-in or have been prescribed by you, displaying the result as output. The input/output alternation continues until you quit the session, with all steps recorded in the cells of your notebook. The simplest expressions involve ordinary numbers (e.g. 1, 2, 3, . , 1/2, 2/3, . , 3.14159), and the familiar operations and relations of arithmetic and logic. Elementary numerical operations: + (plus), - (minus), * (times), / (divided by), ^ (to the power) Elementary numerical relations: == (is equal to), != (is not equal to), < (is less than), > (is greater than), <= (is less than or equal to), >= (is greater than or equal to) Elementary logical relations: || (or), && (and), ! (not) For example, 2+2 evaluates to 4 , 1<2 evaluates to True , and !(2>1) evaluates to False . -
ON the GENESIS of BBP FORMULAS 3 Where R0 and R1 Are Rational Numbers
ON THE GENESIS OF BBP FORMULAS DANIEL BARSKY, VICENTE MUNOZ,˜ AND RICARDO PEREZ-MARCO´ Abstract. We present a general procedure to generate infinitely many BBP and BBP-like formu- las for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for π. We can under- stand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing 0 come from. We also explain what is the observed re- lation between some BBP formulas for log 2 and π, that are obtained by taking real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transal- gebraic considerations, and offer a new way to obtain and to search many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants. 1. Introduction More than 20 years ago, D.H. Bailey, P. Bowein and S. Plouffe ([5]) presented an efficient algorithm to compute deep binary or hexadecimal digits of π without the need to compute the previous ones. Their algorithm is based on a series representation for π given by a formula discovered by S. Plouffe, +∞ 1 4 2 1 1 π = . (1) 16k 8k +1 − 8k +4 − 8k +5 − 8k +6 kX=0 Formulas of similar form for other transcendental constants were known from long time ago, like the classical formula for log 2, that was known to J. Bernoulli, +∞ 1 1 log 2 = . 2k k Xk=1 arXiv:1906.09629v3 [math.NT] 22 Jul 2020 The reader can find in [4] an illustration of the way to extract binary digits from this type of formulae. -
Little Q-Legendre Polynomials and Irrationality of Certain Lambert Series
Little q-Legendre polynomials and irrationality of certain Lambert series Walter Van Assche ∗ Katholieke Universiteit Leuven and Georgia Institute of Technology October 26, 2018 Abstract Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erd˝os [9]. In 1991–1992 Peter Borwein [4] [5] used Pad´eapproximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Ap´ery’s proof of irrationality of ζ(2) and ζ(3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and V¨a¨an¨anen [8] and recently also by Matala-aho and V¨a¨an¨anen [14] (for lnp(2)). In this paper we show how one can obtain rational approximants for hp(1) and lnp(2) (and many other similar quantities) by Pad´eapproximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and V¨a¨an¨anen for hp(1) and a better upper bound as the one given by Matala-aho and V¨a¨an¨anen for lnp(2). 1 Introduction Most important special functions, in particular hypergeometric functions, have q-exten- arXiv:math/0101187v1 [math.CA] 23 Jan 2001 sions, usually obtained by replacing Pochhammer symbols (a)n = a(a +1) (a + n 1) by their q-analog (a; q) = (1 a)(1 aq)(1 aq2) (1 aqn−1). -
Tombstone. the Tombstone of Ludolph Van Ceulen in Leiden, the Netherlands, Is Engraved with His Amazing 35-Digit Approximation to Pi
Tombstone. The tombstone of Ludolph van Ceulen in Leiden, the Netherlands, is engraved with his amazing 35-digit approximation to pi. Notice that, in keeping with the tradition started by Archimedes, the upper and lower limits are given as fractions rather than decimals. (Photo courtesy of Karen Aardal. c Karen Aardal. All rights reserved.) 28 What’s Happening in the Mathematical Sciences Digits of Pi BarryCipra he number π—the ratio of any circle’s circumference to its diameter, or, if you like, the ratio of its area to In 2002, Yasumasa Tthe square of its radius—has fascinated mathematicians Kanada and a team of for millennia. Its decimal expansion, 3.14159265 ..., has been computer scientists at studied for hundreds of years. More recently its binary expan- the University of Tokyo sion, 11.001001 ..., has come under scrutiny. Over the last decade, number theorists have discovered some surprising computed a record new algorithms for computing digits of pi, together with new 1.2411 trillion decimal theorems regarding the apparent randomness of those digits. digits of π,breaking In 2002, Yasumasa Kanada and a team of computer scien- tists at the University of Tokyo computed a record 1.2411 tril- their own previous lion decimal digits of π, breaking their own previous record record of 206 billion of 206 billion digits, set in 1999. The calculation was a shake- digits, set in 1999. down cruise of sorts for a new supercomputer. That’s one of the main reasons for undertaking such massive computations: Calculating π to high accuracy requires a range of numerical methods, such as the fast Fourier transform, that test a com- puter’s ability to store, retrieve, and manipulate large amounts of data. -
This Is a Collection of Mathematical Constants Evaluated to Many Digits
This is a collection of mathematical constants evaluated to many digits. by Simon Plouffe 1995. ----------------------------------------------------------------------------- Contents -------- 1-6/(Pi^2) to 5000 digits. 1/log(2) the inverse of the natural logarithm of 2 to 2000 places. 1/sqrt(2*Pi) to 1024 digits. sum(1/2^(2^n),n=0..infinity). to 1024 digits. 3/(Pi*Pi) to 2000 digits. arctan(1/2) to 1000 digits. The Artin's Constant = product(1-1/(p**2-p),p=prime) The Backhouse constant The Berstein Constant The Catalan Constant The Champernowne Constant Copeland-Erdos constant cos(1) to 15000 digits. The cube root of 3 to 2000 places. 2**(1/3) to 2000 places Zeta(1,2) ot the derivative of Zeta function at 2. The Dubois-Raymond constant exp(1/e) to 2000 places. Gompertz (1825) constant exp(2) to 5000 digits. exp(E) to 2000 places. exp(-1)**exp(-1) to 2000 digits. The exp(gamma) to 1024 places. exp(-exp(1)) to 1024 digits. exp(-gamma) to 500 digits. exp(-1) = exp(Pi) to 5000 digits. exp(-Pi/2) also i**i to 2000 digits. exp(Pi/4) to 2000 digits. exp(Pi)-Pi to 2000 digits. exp(Pi)/Pi**E to 1100 places. Feigenbaum reduction parameter Feigenbaum bifurcation velocity constant Fransen-Robinson constant. gamma or Euler constant GAMMA(1/3) to 256 digits. GAMMA(1/4) to 512 digits. The Euler constant squared to 2000 digits. GAMMA(2/3) to 256 places gamma cubed. to 1024 digits. GAMMA(3/4) to 256 places. gamma**(exp(1) to 1024 digits.