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Divergence Of The Harmonic Series
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: \sum_^\infty\frac = 1 + \frac + \frac + \frac + \frac + \cdots. The first n terms of the series sum to approximately \ln n + \gamma, where \ln is the natural logarithm and \gamma\approx0.577 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. Applications of the harmonic series and its partial sums include Euler's proof that there are infinitely many prime numbers, the analysis of the coupon collector's problem on how many random trials are needed to provide a complete range of responses, the co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quicksort
Quicksort is an efficient, general-purpose sorting algorithm. Quicksort was developed by British computer scientist Tony Hoare in 1959 and published in 1961, it is still a commonly used algorithm for sorting. Overall, it is slightly faster than merge sort and heapsort for randomized data, particularly on larger distributions. Quicksort is a divide-and-conquer algorithm. It works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. For this reason, it is sometimes called partition-exchange sort. The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting. Quicksort is a comparison sort, meaning that it can sort items of any type for which a "less-than" relation (formally, a total order) is defined. Most implementations of quicksort are not stable, meaning that the relative order of equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pietro Mengoli
Pietro Mengoli (1626, Bologna – June 7, 1686, Bologna) was an Italian mathematician and clergyman from Bologna, where he studied with Bonaventura Cavalieri at the University of Bologna, and succeeded him in 1647. He remained as professor there for the next 39 years of his life. Contributions Mengoli first posed the famous Basel problem in 1650, solved in 1735 by Leonhard Euler. In 1650, he also proved that the sum of the alternating harmonic series is equal to the natural logarithm of 2. He also proved that the harmonic series has no upper bound, and provided a proof that Wallis' product for \pi is correct. Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in ''Geometria speciose elementa'' (1659). He used the term ''quasi-infinite'' for unbounded and ''quasi-null'' for vanishing. :Mengoli proves theorems starting from clear hypotheses and explicitly stated properties, showing everything necessary ... proceeds to a step-by-step d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Series
In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar^2 + ar^3 + ..., where a is the coefficient of each term and r is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to frequently used mathematical tools such as the Taylor series, the complex Fourier series, and the matrix exponential. The name geometric series indicates each term is the geometric mean of its two neighboring terms, similar to how the name arithmetic series indicates each term is the arithmetic mean of its two neighboring terms. The sequence of geometric series term ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Swineshead
Richard Swineshead (also Suisset, Suiseth, etc.; fl. c. 1340 – 1354) was an English mathematician, logician, and natural philosopher. He was perhaps the greatest of the Oxford Calculators of Merton College, where he was a fellow certainly by 1344 and possibly by 1340. His magnum opus was a series of treatises known as the ''Liber calculationum'' ("Book of Calculations"), written c. 1350, which earned him the nickname of The Calculator. Robert Burton (d. 1640) wrote in ''The Anatomy of Melancholy'' that "Scaliger and Cardan admire Suisset the calculator, ''qui pene modum excessit humani ingenii'' hose talents were almost superhuman. Gottfried Leibniz wrote in a letter of 1714: "Il y a eu autrefois un Suisse, qui avoit mathématisé dans la Scholastique: ses Ouvrages sont peu connus; mais ce que j'en ai vu m'a paru profond et considérable." ("There was once a Suisse, who did mathematics belonging to scholasticism; his works are little known, but what I have seen of them seeme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proportion (architecture)
Proportion is a central principle of architectural theory and an important connection between mathematics and art. It is the visual effect of the relationships of the various objects and spaces that make up a structure to one another and to the whole. These relationships are often governed by multiples of a standard unit of length known as a "module". Proportion in architecture was discussed by Vitruvius, Leon Battista Alberti, Andrea Palladio, and Le Corbusier among others. Roman architecture Vitruvius Architecture in Roman antiquity was rarely documented except in the writings of Vitruvius' treatise '' De architectura''. Vitruvius served as an engineer under Julius Caesar during the first Gallic Wars (58–50 BC). The treatise was dedicated to Emperor Augustus. As Vitruvius defined the concept in the first chapters of the treatise, he mentioned the three prerequisites of architecture are firmness (''firmitas''), commodity (''utilitas''), and delight (''venustas''), whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Baroque
The Baroque (, ; ) is a style of architecture, music, dance, painting, sculpture, poetry, and other arts that flourished in Europe from the early 17th century until the 1750s. In the territories of the Spanish and Portuguese empires including the Iberian Peninsula it continued, together with new styles, until the first decade of the 19th century. It followed Renaissance art and Mannerism and preceded the Rococo (in the past often referred to as "late Baroque") and Neoclassical styles. It was encouraged by the Catholic Church as a means to counter the simplicity and austerity of Protestant architecture, art, and music, though Lutheran Baroque art developed in parts of Europe as well. The Baroque style used contrast, movement, exuberant detail, deep colour, grandeur, and surprise to achieve a sense of awe. The style began at the start of the 17th century in Rome, then spread rapidly to France, northern Italy, Spain, and Portugal, then to Austria, southern Germany, and Russia. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Progression (mathematics)
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form : \frac,\ \frac,\ \frac,\ \frac, \cdots, where ''a'' is not zero and −''a''/''d'' is not a natural number, or a finite sequence of the form : \frac,\ \frac,\ \frac,\ \frac, \cdots,\ \frac, where ''a'' is not zero, ''k'' is a natural number and −''a''/''d'' is not a natural number or is greater than ''k''. Examples * 1, 1/2, 1/3, 1/4, 1/5, 1/6, sometimes referred to as the ''harmonic sequence'' * 12, 6, 4, 3, \tfrac, 2, … , \tfrac, … * 30, −30, −10, −6, − \tfrac, … , \tfrac * 10, 30, −30, −10, −6, − , … , \tfrac Sums of harmonic progressions Infinite harmonic progressions are not summable (sum to infinity). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is : \left(\frac\right)^ = \frac = \frac = 2\,. Definition The harmonic mean ''H'' of the positive real numbers x_1, x_2, \ldots, x_n is defined to be :H = \frac = \frac = \left(\frac\right)^. The third formula in the above equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals. From the following formula: :H = \frac. it is more apparent that the harmonic mean is related to the arithmetic and geometric means. It is the reciprocal dual of the arithmetic mean for positive inputs: :1/H(1/x_1 \ldots 1/x_n) = A(x_1 \ldots x_n) The harmonic mean is a Schur-con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Frequency
The fundamental frequency, often referred to simply as the ''fundamental'', is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as 0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as 1, the first harmonic. (The second harmonic is then 2 = 2⋅1, etc. In this context, the zeroth harmonic would be 0 Hz.) According to Benward's and Saker's ''Music: In Theory and Practice'': Explanation All sinusoidal and many non-sinusoidal waveforms repeat exactly over time – they are periodic. The period of a waveform is the smallest value of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter ''lambda'' (λ). The term ''wavelength'' is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. Wavelength depends on the medium (for example, vacuum, air, or water) that a wav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
