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A triple of positive integers satisfying is said to be harmonic ifIn particular, such a triple is harmonic if the reciprocals of its terms form an arithmetic sequence with common difference whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of harmonic triples and the set of equivalence classes of geometric triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .

A triple of positive integers satisfying is said to be geometric if . In particular, such a triple is geometric if its terms form a geometric sequence with common ratio whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of geometric triples and the set of equivalence classes of harmonic triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .

The Cookson Hills series is the series similar to the Flint Hills series, but with numerator instead of :(Pickover 2002, p. 268). It is not known if this series converges since can have sporadic large values. The plots above show its behavior up to . The positive integer values of giving incrementally largest values of are given by 1, 2, 5, 8, 11, 344, 699, 1054, 1409, 1764, 2119, ... (OEIS A004112), corresponding to the values 1.85082, 2.403, 3.52532, 6.87285, 225.953, 227.503, ....

A Tauberian theorem is a theorem that deduces the convergence of an series on the basis of the properties of the function it defines and any kind of auxiliary hypothesis which prevents the general term of the series from converging to zero too slowly. Hardy (1999, p. 46) states that "a 'Tauberian' theorem may be defined as a corrected form of the false converse of an 'Abelian theorem.' "Wiener's Tauberian theorem states that if , then the translates of span a dense subspace iff the Fourier transform is nonzero everywhere. This theorem is analogous with the theorem that if (for a Banach algebra with a unit), then spans the whole space if and only if the Gelfand transform is nonzero everywhere.

Like the entire harmonic series, the harmonicseries(1)taken over all primes also diverges, as first shown by Euler in 1737 (Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil 2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3, ... after 3, 59, 361139, ... (OEIS A046024) primes.Its asymptotic behavior is given by(2)where(3)(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003, p. 64).

The Cauchy remainder is a different form of the remainder term than the Lagrange remainder. The Cauchy remainder after terms of the Taylor series for a function expanded about a point is given bywhere (Hamilton 1952).Note that the Cauchy remainder is also sometimes taken to refer to the remainder when terms up to the st power are taken in the Taylor series, and that a notation in which , , and is sometimes used (Blumenthal 1926; Whittaker and Watson 1990, pp. 95-96).

If(1)then(2)is given by(3)where .When applied to the generating function(4)it gives the identity(5)with integers (and where the sum can be taken only up to ).Other multisection examples are given by Somos (2006).

The series(1)is called the harmonic series. It can be shown to diverge using the integral test by comparison with the function . The divergence, however, is very slow. Divergence of the harmonic series was first demonstrated by Nicole d'Oresme (ca. 1323-1382), but was mislaid for several centuries (Havil 2003, p. 23; Derbyshire 2004, pp. 9-10). The result was proved again by Pietro Mengoli in 1647, by Johann Bernoulli in 1687, and by Jakob Bernoulli shortly thereafter (Derbyshire 2004, pp. 9-10).Progressions of the form(2)are also sometimes called harmonic series (Beyer 1987).Oresme's proof groups the harmonic terms by taking 2, 4, 8, 16, ... terms (after the first two) and noting that each such block has a sum larger than 1/2,(3)(4)and since an infinite sum of 1/2's diverges, so does the harmonic series.The generalization of the harmonic series(5)is known as the Riemann zeta function.The sum of the first few terms of..

The bias of a series is defined asA series is geometric iff . A series is artistic iff the bias is constant.

Multisection of a mathematical quantity or figure is division of it into a number of (usually) equal parts. Division of a quantity into two equal parts is known as bisection, and into three equal parts is known as trisection.The coordinates of the first -multisection of a line segment with endpoints given in trilinear coordinates by and is , where(1)(2)(3)

How far can a stack of books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible for books (in terms of book lengths) is half the th partial sum of the harmonic series.This is given explicitly by(1)where is a harmonic number. The first few values are(2)(3)(4)(5)(OEIS A001008 and A002805).When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is(6)(7)(8)(Derbyshire 2004, p. 6).In order to find the number of stacked books required to obtain book-lengths of overhang, solve the equation for , and take the ceiling function. For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.When more than one book or card can be used per level, the problem becomes much more complex. For..

A series is an infinite ordered set of terms combined together by the addition operator. The term "infinite series" is sometimes used to emphasize the fact that series contain an infinite number of terms. The order of the terms in a series can matter, since the Riemann series theorem states that, by a suitable rearrangement of terms, a so-called conditionally convergent series may be made to converge to any desired value, or to diverge.Conditions for convergence of a series can be determined in the Wolfram Language using SumConvergence[a, n].If the difference between successive terms of a series is a constant, then the series is said to be an arithmetic series. A series for which the ratio of each two consecutive terms is a constant function of the summation index is called a geometric series. The more general case of the ratio a rational function of produces a series called a hypergeometric series.A series may converge to a definite value,..

A Taylor series remainder formula that gives after terms of the seriesfor and any (Blumenthal 1926, Beesack 1966), which Blumenthal (1926) ascribes to Roche (1858). The choices and give the Lagrange and Cauchy remainders, respectively (Beesack 1966).

The identity(1)holds for any integrable function and of the form(2)with , , and arbitrary constants (Glasser 1983). Here, denotes a Cauchy principal value. This generalized the result known to Cauchy that(3)where .

There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the "binomial series" rather than "binomial theorem."The most general case of the binomial theorem is the binomialseries identity(1)where is a binomial coefficient and is a real number. This series converges for an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with the binomial theorem.When is a positive integer , the series terminates at and can be written in the form(2)This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).The differing terminologies are..

The Rayleigh functions for , 2, ..., are defined aswhere are the zeros of the Bessel function of the first kind (Watson 1966, p. 502; Gupta and Muldoon 1999). They were used by Euler, Rayleigh, and others to evaluate zeros of Bessel functions.There is a convolution formula connecting Rayleigh functions of different orders,(Kishore 1963, Gupta and Muldoon 1999).

There are several related series that are known as the binomial series.The most general is(1)where is a binomial coefficient and is a real number. This series converges for an integer, or (Graham et al. 1994, p. 162). When is a positive integer , the series terminates at and can be written in the form(2)The theorem that any one of these (or several other related forms) holds is knownas the binomial theorem.Special cases give the Taylor series(3)(4)where is a Pochhammer symbol and . Similarly,(5)(6)which is the so-called negative binomial series.In particular, the case gives(7)(8)(9)(OEIS A001790 and A046161), where is a double factorial and is a binomial coefficient.The binomial series has the continued fractionrepresentation(10)(Wall 1948, p. 343).

A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation index . The more general case of the ratio a rational function of the summation index produces a series called a hypergeometric series.For the simplest case of the ratio equal to a constant , the terms are of the form . Letting , the geometric sequence with constant is given by(1)is given by(2)Multiplying both sides by gives(3)and subtracting (3) from (2) then gives(4)(5)so(6)For , the sum converges as ,in which case(7)Similarly, if the sums are taken starting at instead of ,(8)(9)the latter of which is valid for .

The series which arises in the binomial theorem for negative integer ,(1)(2)for .For , the negative binomial series simplifies to(3)

The Flint Hills series is the series(Pickover 2002, p. 59). It is not known if this series converges, since can have sporadic large values. The plots above show its behavior up to . The positive integer values of giving incrementally largest values of are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of the convergents of , corresponding to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....Alekseyev (2011) has shown that the question of the convergence of the Flint Hill series is related to the irrationality measure of , and in particular, convergence would imply , which is much stronger than the best currently known upper bound.

A series of the form(1)where is a real and is a Bessel function of the first kind. Special cases are(2)where is the gamma function, and(3)where(4)and is the floor function.

The exponential sum function , sometimes also denoted , is defined by(1)(2)where is the upper incomplete gamma function and is the (complete) gamma function.

A multinomial series is generalization of the binomial series discovered by Johann Bernoulli and Leibniz. The multinomial series arises in a generalization of the binomial distribution called the multinomial distribution.It is given bywhere .For example,

An arithmetic series is the sum of a sequence , , 2, ..., in which each term is computed from the previous one by adding (or subtracting) a constant . Therefore, for ,(1)The sum of the sequence of the first terms is then given by(2)(3)(4)(5)(6)Using the sum identity(7)then gives(8)Note, however, that(9)so(10)or times the arithmetic mean of the first and last terms! This is the trick Gauss used as a schoolboy to solve the problem of summing the integers from 1 to 100 given as busy-work by his teacher. While his classmates toiled away doing the addition longhand, Gauss wrote a single number, the correct answer(11)on his slate (Burton 1989, pp. 80-81; Hoffman 1998, p. 207). When the answers were examined, Gauss's proved to be the only correct one.

A Lambert series is a series of the form(1)for . Then(2)(3)where(4)The particular case is sometimes denoted(5)(6)(7)for (Borwein and Borwein 1987, pp. 91 and 95), where is a q-polygamma function. Special cases and related sums include(8)(9)(10)(11)(12)(Borwein and Borwein 1997, pp. 91-92), which arise in the reciprocalFibonacci and reciprocal Lucas constants.Some beautiful series of this type include(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)where is the Möbius function, is the totient function, is the number of divisors of , is the q-polygamma function, is the divisor function, is the number of representations of in the form where and are rational integers (Hardy and Wright 1979), is a Jacobi elliptic function (Bailey et al. 2006), is the Liouville function, and is the least significant bit of ...

An arithmetic progression, also known as an arithmetic sequence, is a sequence of numbers such that the differences between successive terms is a constant .An arithmetic progression can be generated in the Wolfram Language using the command Range[, , d].

A Kapteyn series is a series of the form(1)where is a Bessel function of the first kind. Examples include Kapteyn's original series(2)and(3)

A series of the form(1)or(2)where .A series with positive terms can be converted to an alternating series using(3)where(4)Explicit values for alternating series include(5)(6)(7)(8)where is Apéry's constant, and sums of the form (6) through (8) are special cases of the Dirichlet eta function.The following alternating series converges, but a closed form is apparently not known,(9)(10)(11)(OEIS A114884).

The "Foxtrot series" is a mathematical sum that appeared in the June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007). It arose from a convergence testing problem in a calculus book by Anton, but was inadvertently converted into a summation problem on an alleged final exam by the strip's author:(1)The sum can be done using partial fraction decomposition to obtain(2)(3)(4)(5)(OEIS A127198), where and the last sums have been done in terms of the digamma function and symbolically simplified.

Series reversion is the computation of the coefficients of the inverse function given those of the forward function. For a function expressed in a series with no constant term (i.e., ) as(1)the series expansion of the inverse series is given by(2)By plugging (2) into (1), the following equationis obtained(3)Equating coefficients then gives(4)(5)(6)(7)(8)(9)(10)(Dwight 1961, Abramowitz and Stegun 1972, p. 16).Series reversion is implemented in the Wolfram Language as InverseSeries[s, x], where is given as a SeriesData object. For example, to obtain the terms shown above, With[{n = 7}, CoefficientList[ InverseSeries[SeriesData[x, 0, Array[a, n], 1, n + 1, 1]], x] ]A derivation of the explicit formula for the th term is given by Morse and Feshbach (1953),(11)where(12)

In response to a letter from Goldbach, Euler considered sums ofthe form(1)(2)with and and where is the Euler-Mascheroni constant and is the digamma function. Euler found explicit formulas in terms of the Riemann zeta function for with , and E. Au-Yeung numerically discovered(3)where is the Riemann zeta function, which was subsequently rigorously proven true (Borwein and Borwein 1995). Sums involving can be re-expressed in terms of sums the form via(4)(5)(6)and(7)where is defined below.Bailey et al. (1994) subsequently considered sums ofthe forms(8)(9)(10)(11)(12)(13)(14)(15)where and have the special forms(16)(17)(18)where is a generalized harmonic number.A number of these sums can be expressed in terms of the multivariatezeta function, e.g.,(19)(Bailey et al. 2006a, p. 39, sign corrected; Bailey et al. 2006b).Special cases include(20)(P. Simone, pers. comm., Aug. 30, 2004).Analytic single..

The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging into the Leibniz series,(1)(Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular(2)where is the Riemann zeta function (Vardi 1991).Taking the partial series gives the analytic result(3)Rather amazingly, expanding about infinity gives the series(4)(Borwein and Bailey 2003, p. 50), where is an Euler number. This means that truncating the Gregory series at half a large power of 10 can give a decimal expansion for whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking gives where the sequence of differences is precisely twice the Euler (secant) numbers. In fact, just this pattern of digits was observed by J. R. North in 1988 before the closed form of the truncated series was known..

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