The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

There are also symbols for series of chemical elements, for comparative formulas.

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

This has the advantage of simplifying many of the formulas involved, and provides a formulation for Fourier series that more closely resembles the definition followed in this article.

II. 12 ) the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers.

The novels relied on a series of predictable literary formulas appealing to mass tastes and were often written in as little as a few days.

In the theory of power series, it is also often treated as 1 by convention, to make certain formulas more concise.

During the 1890s Bukreev published a series of high quality papers including: " On the theory of gamma functions ," " On some formulas in the theory of elliptic functions of Weierstrass ," " On the distribution of the roots of a class of entire transcendental functions ," and " Theorems for elliptic functions of Weierstrass ".

This approximate formula is for moderate to large sample sizes ; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes.

When f is a set of non-linear combination of the variables x, it must usually be linearized by approximation to a first-order Taylor series expansion, though in some cases, exact formulas can be derived that do not depend on the expansion as is the case for the exact variance of products.

With the allowing Don Panoz to bring the rules and racing formulas of the 24 Hours of Le Mans to North America with the creation of the Petit Le Mans in 1998, Don Panoz attempted to build up a series to be based around Petit Le Mans.

In the Flemish Suske en Wiske series, on one occasion a thought balloon full of mathematical formulas is cut open with scissors and its contents emptied in a bag, to be saved for later ( in a manner not unlike the pensieve in the Harry Potter series ).

The result is that the formulas for the parameters of each variable's distributions can be expressed as a series of equations with mutual, nonlinear dependencies among the variables.

Similar techniques involving holomorphic Eisenstein series twisted by a Dirichlet character produce formulas for the number of representations of a positive integer n as a sum of two, four, or eight squares in terms of the divisors of n.

These formulas are truncated version of Transverse Mercator: flattening series, which were originally derived by Johann Heinrich Louis Krüger in 1912.

Their kind created an American culture superior to any in the world, an industrial and technological culture which penetrated Russia as it did almost every corner of the earth without a nickel from the Federal treasury or a single governmental specialist to contrive directives or program a series of consultations of interested agencies.

According to The Chicago Tribune News Service, State Atty. Gen. Stanley Mosk of California has devised a series of questions which the joiner might well ask about any organization seeking his money and his name: 1.

This is a radical change in attitude from the conditions which prevailed several years ago, when a series of bombings was directed against Negroes who were moving into previously all-white neighborhoods of Dallas.

Among some recent imports were seat covers for one series of dining room chairs on which were depicted salad plates overflowing with tomatoes and greens and another set on which a pineapple was worked in naturalistic color.

RCA Victor has an ambitious and useful project in a stereo series called `` Adventures In Music '', which is an instructional record library for elementary schools.

The second great dilemma has been the morality of nuclear testing, a dilemma which has suddenly become acute because of the present series of Soviet tests.

But I have compared its text with already published commentaries on the 1960 series of Godkin lectures at Harvard, from which the book was derived, and I can with confidence challenge the gist of C. P. Snow's incautious tale ''.

Below these particulars was a series of alpha-beta-gammas connected by arrows and crosses which denoted the lineage of the breed.

-- Although there was some variation in results which must be attributed either to trace impurities or to variation in wall effects, the photochemical exchange in the gas phase was sufficiently reproducible so that it seemed meaningful to compare the reaction rates in different series of reaction tubes for the purpose of obtaining information on the effect of chlorine concentration and of carbon tetrachloride concentration on the reaction rate.

Thus in Mary wrote an account of the trip first strong stress on Mary marks Mary as the first in a series of people who wrote accounts of the trip, strong stress on wrote marks the writing as the first of a series of actions of Mary's concerned with an account of her trip ( about which she may later have made speeches, for example ), and strong stress on trip makes the trip the first of a series of subjects about which Mary wrote accounts.

These two recollections form the frame around a series of experiences and sights which, to me at least, symbolize the extremes in the aesthetic as well as ethical conflict between materialism and humanism.

Instead, we became involved in a series of friendly, but overly formal, welcoming addresses to which we had no choice but to reply in kind.

It consists of a series of pipes and a pressure-measuring chamber which record the rise and fall of the water surface.

Taylor said Mrs. Huntley and her husband also will be questioned about a series of 15 Portland robberies in spring of 1959 in which the holdup men bound their victims with tape before fleeing.

The Leningrad Kirov Ballet, which opened a series of performances Friday night at the Opera House, is, I think, the finest `` classical '' ballet company I have ever seen, and the production of the Petipa-Tschaikowsky `` Sleeping Beauty '' with which it began the series is incomparably the finest I have ever had the pleasure of witnessing.

However, the wei books were also destroyed in a series of Orthodox Confucian purges which culminated in a final proscription in 605.

Let us quote once more from R. G. Collingwood: `` History is properly concerned with the actions of human beings Regarded from the outside, an action is an event or series of events occurring in the physical world ; ;

Pauling's estimate of 200 megatons yield from the present series of Russian tests will probably turn out to be too high, but a total of 100 megatons is a distinct possibility.

Secondly, a whole series of addresses and actions by the Pope and by others show that concern for Christian unity is still very much alive and growing within the Church.

Using Edgerton's method, the fast-closing action is obtained from the blackening of a window by exploding a series of parallel lead wires.

The exercise I shall discuss in this -- the first of a new series of articles on muscle definition-specialization of a particular body part -- is the One Leg Lunge.

A tsunami is not a single wave but a series.

Hypothesizing a series of developmental stages that begin in the individual's infancy and end in his old age, Erikson has indicated that the adolescent is faced with a series of identity crises.

In a series of fairy tales and fantasies, Melies demonstrated that the film is superbly equipped to tell a straightforward story, with beginning, middle and end, complications, resolutions, climaxes, and conclusions.

Physically, a movie is possible because a series of images is projected one at a time at such a speed that the eye `` remembers '' the one that has gone before even as it registers the one now appearing.

Perhaps the Pirate who will be the unhappiest over the news that Musial probably will sit out most of the series is Bob Friend, who was beaten by The Man twice last season on dramatic home runs.

The series of ballets is sponsored by the Milenoff Ballet Foundation, Inc., a non-profit foundation with headquarters in Coral Gables.

About that same time John Crosby's TV series on the popular arts proved again that giving jazz ample breathing space is one of the most sensible things a producer can do.

London explains that the very distinct directional effect in the Phase 4 series is due in large part to their novel methods of microphoning and recording the music on a number of separate tape channels.

This is not only a compliment to Mijbil, of whom there are a fine series of photographs and drawings in the book, but to the author who has catalogued the saga of a frightened otter cub's journey by plane from Iraq to London, then by train ( where he lay curled in the wash basin playing with the water tap ) to Camusfearna, with affectionate detail.

This music is characterized by a large technical research and focuses mainly on twelve long Noubate " series ", its main instruments are the mandolin, violin, lute, guitar, zither, flute and piano.

A series of linked carbon atoms is known as the carbon skeleton or carbon backbone.