A proof by contradiction follows. Assume that the sum of the reciprocals of the primes converges: Define pi as the ith prime number. We have: There exists a positive integer i such that: Define N(x) as the number of positive integers n not exceeding x and not divisible by a prime other than the first i ones. Let us write this n as kmSee also:

Proof that the sum of the reciprocals of the primes diverges, Proof that the sum of the reciprocals of the primes diverges - The harmonic series, Proof that the sum of the reciprocals of the primes diverges - First proof, Proof that the sum of the reciprocals of the primes diverges - Second proof, Proof that the sum of the reciprocals of the primes diverges - Third proof, Proof that the sum of the reciprocals of the primes diverges - External link

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Titles in alphabetical order. Note: Some depictions were made before the space elevator concept became known. Space elevator - Novels and Fairy tales. 3001: The Final Odyssey, novel by Arthur C. Clarke Assassin Gambit, novel by William Forstchen. Chasm City, a novel by Alastair Reynolds Feersum Endjinn, novel by Iain M. Banks Foreigner, novel by Robert J. Sawyer Friday, novel by Robert A. Heinlein Hothouse, novel by Brian Aldiss. J ...

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Space elevator, Space elevator - Orbital tethers, Space elevator - Physics and structure, Space elevator - Base station, Space elevator - Cable, Space elevator - Climbers, Space elevator - Counterweight, Space elevator - Launching into outer space, Space elevator - Extraterrestrial elevators, Space elevator - Construction, Space elevator - Traditional way, Space elevator - Brad Edwards' proposal, Space elevator - Other designs, Space elevator - Failure modes and safety issues, Space elevator - Satellites, Space elevator - Meteoroids and micrometeorites, Space elevator - Corrosion, Space elevator - Weather, Space elevator - Sabotage, Space elevator - Vibrational harmonics, Space elevator - In the event of failure, Space elevator - Van Allen Belts, Space elevator - Economics, Space elevator - Political issues, Space elevator - History, Space elevator - Fiction, Space elevator - Novels and Fairy tales, Space elevator - Anime Comics and Manga, Space elevator - Videogames, Space elevator - Movies and TV series, Space elevator - Others

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In most musical instruments, the tone-generating component (a string or resonant column of air) vibrates at multiple frequencies simultaneously: a fundamental frequency that is usually perceived as the pitch of the note, and harmonics or overtones that are multiples of the fundamental frequency and whose wavelengths therefore divide the tone-generating region into simple fractional segments (1/2, 1/3, 1/4, etc.). (See harmonic series.) The fundamental note and its harmonics sound together, and the amplitude relationships among them ...

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Stretched tuning, Stretched tuning - Fundamentals and harmonics, Stretched tuning - Intervals and inharmonicity, Stretched tuning - Vibration of wire strings, Stretched tuning - Tines and reeds, Stretched tuning - Effects on tuning, Stretched tuning - References and further information

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As noted above, valves allow brass players to change pitches A piston valve is a device used to change the pitch of a brass instrument; three or more piston valves can be found on trumpets, tubas, and the like. When opened ("pressed" and "pushed down"), each valve changes the pitch by diverting the air stream through additional tubing, thus lengthening the instrument and lowering the harmonic series on which the instrument is vibrating. The following list shows how each valve or combination of valves will affect the pitch fr ...

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Brass instrument, Brass instrument - Families of brass instruments, Brass instrument - Some other wind instruments, Brass instrument - Valves, Brass instrument - Sound production in brass instruments

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Let f(x) = x be the identity function for x from −π to π. Outside this domain, the Fourier series implicitly requires that we define the function periodically. We will compute the Fourier coefficients for this function. Notice that cos(nx) is an even function, while f and sin(nx) are odd functions. Notice that an are 0 because x and x cos(nx) are odd functions. Hence the Fourier series for f(x) = x is: See also:

Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

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While the Fourier coefficients an and bn can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f. The simplest answer is that if f is square-integrable then (this is convergence in the norm of the space L2). There are also many known tests that ensure that the series converges at a given poin ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

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The Fourier basis functions are orthogonal in the discrete space where δ(x) is the Dirac delta function and δT(x) is the Dirac comb function. The Fourier basis functions are orthogonal in the continuous space as well: where δnm is the Kronecker delta function. ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

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Because "basis functions" eikx are homomorphisms of the real line (more precisely, of the "circle group") we have some useful identities: Fourier series - Shifting property. If then (if G is the transform of g) Fourier series - Convolution theorems. Main article: Convolution If h( ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

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The very earliest amplifiers usually had single-ended topologies with the most basic type of vacuum tube, known as a triode. An audio amplifier using this topology will always be in class A. Class A single-ended triode amplifiers (known as SETs) have a characteristic asymmetrical distortion spectrum, a simple and monotonically decaying series of harmonics, dominated by modest levels of second harmonic distortion and followed by both even- and odd-numbered harmonics. Second harmonic distortion (multiplication of the original frequencie ...

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Valve sound, Valve sound - Audible differences, Valve sound - Psychoacoustics, Valve sound - Explanation, Valve sound - Device characteristics and distortion, Valve sound - Modern amplifier choices, Valve sound - Amplifier 'class', Valve sound - Amplifier bandwidth, Valve sound - Asymmetry, Valve sound - Negative feedback, Valve sound - Power supplies, Valve sound - Signal source limitations, Valve sound - Valve sound from transistor amplifiers, Valve sound - Transistor sound from valve amplifiers, Valve sound - Intentional distortion, Valve sound - Tubes, Valve sound - Solid state soft limiters, Valve sound - Valve sound trivia

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Another important property of the Fourier series is the Plancherel theorem Parseval's theorem, a special case of the Plancherel theorem, states that which can be restated for the real-valued f(x) case above, These theorems may be proven using the orthogonality relationships. ...

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Fourier series, Fourier series - Definition of Fourier series, Fourier series - Example, Fourier series - Convergence of Fourier series, Fourier series - Orthogonality, Fourier series - Some positive consequences of the homomorphism properties of exp, Fourier series - Shifting property, Fourier series - Convolution theorems, Fourier series - Plancherel's and Parseval's theorem, Fourier series - General formulation

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Consider two bodies (not affected by gravity), each of mass M, attached to three springs with stiffness K. They are attached in the following manner: where the edge points are fixed and cannot move. We'll use x1(t) to denote the displacement of the leftmost mass, and x2(t) to denote the displacement of the rightmost. If we denote the second derivative of x(t) with respect to time as x″, the equations of motion are: See also:

Normal mode, Normal mode - Example - normal modes of coupled oscillators, Normal mode - Standing waves, Normal mode - Normal modes in quantum mechanics

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A standing wave is a continuous form of normal mode. In a standing wave, all the space elements (i.e (x,y,z) coordinates) are oscillating in the same frequency and in phase (reaching the equilibrium point together), but each has a different amplitude. The general form of a standing wave is: Ψ(t) = f(x,y,z)(Acos(ωt) + Bsin(ωt)) where f(x, y, z) represents the dependence of amplitude on location and ...

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Normal mode, Normal mode - Example - normal modes of coupled oscillators, Normal mode - Standing waves, Normal mode - Normal modes in quantum mechanics

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In quantum mechanics, a state of a system is described by a wavefunction of (x, t) which solves the Schrödinger equation. The square of the absolute value of ,i.e. is the probability (density) to measure the particle in place x at time t. Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscill ...

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Normal mode, Normal mode - Example - normal modes of coupled oscillators, Normal mode - Standing waves, Normal mode - Normal modes in quantum mechanics

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ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if exists where N(A, n) is the number of members of A less than or equal to n, then is equal to that density. The latter limit can also exist in some cases in which A does not have a density. For example, if A is the set of all positive integers whose first ...

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Zeta distribution, Zeta distribution - Moments, Zeta distribution - Moment generating function, Zeta distribution - The case s = 1

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Sonic the Hedgehog TV series - Season 1. Heads or Tails Sonic Past Cool Sub Sonic Warp Sonic Sonic and Sally Ultra Sonic Sonic Racer Hooked on Sonics Harmonic Sonic Sonic's Nightmare Sonic Boom Super Sonic Sonic and the Secret Scrolls Sonic the Hedgehog TV series - Season 2. Game Guy Sonic Conversion No Brainer Blast to the Pa ...

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Sonic the Hedgehog TV series, Sonic the Hedgehog TV series - Plot Summary, Sonic the Hedgehog TV series - Theme Song, Sonic the Hedgehog TV series - Vocal Talents, Sonic the Hedgehog TV series - Episode List, Sonic the Hedgehog TV series - Season 1, Sonic the Hedgehog TV series - Season 2, Sonic the Hedgehog TV series - Season 3 Aborted

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Each note produced by a musical instrument is made of a number of distinct frequencies, measured in hertz (Hz). The lowest frequency is called the fundamental and the pitch produced by this frequency is used to name the note. For example, in western music, instruments are normally tuned to A = 440 Hz. However, the richness of the sound is produced by the combination of this fundamental with a series of harmonics and/or partials (also collectively called overtones). Most western instruments produce harmonic sounds, and these can ...

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Timbre, Timbre - Terms, Timbre - American Standards Association definition, Timbre - Attributes, Timbre - Spectra, Timbre - Envelope, Timbre - In music, Timbre - Sources

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In contrast to the sawtooth wave, which contains all integer harmonics, the square wave contains only odd integer harmonics. Using Fourier series we can write an ideal square wave as an infinite series of the form A curiosity of the convergence of the Fourier series representation of the square wave is the Gibbs phenomenon. Ringing artifacts in non-ideal square waves can be shown to be related to this phenomenon. The Gibbs phenomenon can be prevented by the use of σ-approximation, which uses the Lanczos s ...

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Square wave, Square wave - Origins and uses, Square wave - Examining the square wave, Square wave - Characteristics of imperfect square waves, Square wave - Other definitions

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NCIS TV series - Main Characters. Special Agent Leroy Jethro Gibbs (Mark Harmon) A senior NCIS agent. An experienced former US Marine Gunnery Sergeant and sniper, Agent Gibbs is a highly skilled investigator and interrogator endowed with seemingly infallible intuition which he uses to his advantage while working on a case. While being a man who "gets the job done," it has been frequently shown that Gibbs has little patience with bureaucracy. As of early season three, there have been many references ...

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NCIS TV series, NCIS TV series - Characters, NCIS TV series - Main Characters, NCIS TV series - Recurring Characters, NCIS TV series - Naming the show, NCIS TV series - Trivia, NCIS TV series - Episodes

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Quasar I, II, and (last) M8 (1975-1977) $20,000 USD base price Dual Motorola 6800 CPUs Made by Fairlight and Creative Strategies 8 voices (no sampling, just numeric additive synthesis with 128 harmonics) Memory: 4 kB per voice Synthesis: Fourier synthesis; dynamic harmonic control, waveform editing Hole paper tape reader CMI Series I (1979) 12,000 British pounds The first musical sampler 8 voices ...

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Fairlight CMI, Fairlight CMI - History, Fairlight CMI - Influence, Fairlight CMI - Features Timeline, Fairlight CMI - Sound Clips, Fairlight CMI - Artists using the Fairlight CMI

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Early amplifiers were of necessity valve amplifiers since the transistor did not become common in consumer amplifiers until the late 1960s. The very earliest amplifiers usually had single-ended topologies with the most basic type of vacuum tube, known as a triode. An audio amplifier using this topology will always be in class A. Class A single-ended triode amplifiers (known as SET's) have a characteristic asymmetrical distortion spectrum, a simple and monotonically decaying series of harmonics, dominated by modest levels of second harmonic d ...

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Valve sound, Valve sound - Device characteristics, Valve sound - Bandwidth, Valve sound - Asymmetry, Valve sound - Negative feedback, Valve sound - Power Supplies, Valve sound - Modern amplifiers

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Note: All military ranks given for currently used characters are their ranks as of the end of the series in April 2005. For characters that left the series, ranks given are their last known ranks. The final ensemble cast centres on United States Navy Captain Harmon "Harm" Rabb, Jr., played by David James Elliott, and United States Marine Corps Lieutenant Colonel Sarah "Mac" MacKenzie, played by Catherine Bell. Elliott's character was promoted to Captain in the second-to-last episode of the series. Their obvious attraction to each other, which must not be allowed to interfere with their professional relationship, is ...

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JAG, JAG - Main cast, JAG - Recurring supporting cast, JAG - NCIS spin-off, JAG - Episodes, JAG - JAG around the world, JAG - JAG on DVD

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The distance between each step and the next is aurally the same for any two adjacent steps; though, because steps form a geometric sequence, the difference in frequency increases from one to the next. A linear sequence of one frequency difference would create ever smaller intervals (ratios), such as the harmonic series. See also logarithmic scale. Equal temperaments allow the use of integer notation; a single integer can be used to represent the pitch. The pitch classes can then be expressed in terms of modular arithmetic modulo the number of divisions of the octave, and this expedites mathematic ...

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Equal temperament, Equal temperament - Explanation, Equal temperament - History, Equal temperament - Twelve-tone equal temperament, Equal temperament - Cent values of equal temperament, Equal temperament - Non-12 TET, Equal temperament - Sources

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