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.

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable ways to calculate sum of powers.

The formula used to calculate the final scores is a weighted sum of the votes.

For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence.

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

Euler computed this sum to 20 decimal places with only a few terms of the Euler – Maclaurin formula in 1735.

Thus a conventionally corrected form of Newcomb's formula, to incorporate the corrections on the basis of mean solar time, would be the sum of the two preceding expressions:

Let now x ' and y ' be tuples of previously unused variables of the same length as x and y respectively, and let Q be a previously unused relation symbol which takes as many arguments as the sum of lengths of x and y ; we consider the formula

By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication.

Another reason can be seen by looking at the formula for the kurtosis of the sum of random variables.

In fact, knowing the size of the deck and the size of the cuts, the formula for the composite single cut is given as the sum of the sizes of the cuts modulo the size of the deck.

) In modern mathematics, this formula can be derived using integral calculus, i. e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked centered side by side along the x axis from where the disk has radius r ( i. e. ) to where the disk has radius 0 ( i. e. ).

In the case of the plane ( where the Gaussian curvature is 0 and geodesics are straight lines ), we recover the familiar formula for the sum of angles in an ordinary triangle.

The formula implies that bound systems have an invariant mass ( rest mass for the system ) less than the sum of their parts, if the binding energy has been allowed to escape the system after the system has been bound.

Stirling's formula, without the factor that is often irrelevant in applications, can be quickly obtained by approximating the sum

For example, the annuity formula is the sum of a series of present value calculations.

The formula for the present value of a regular stream of future payments ( an annuity ) is derived from a sum of the formula for future value of a single future payment, as below, where C is the payment amount and n the period.

Four unequal lengths, each less than then sum of the other three, are the sides of each of three non-congruent cyclic quadrilaterals, which by Brahmagupta's formula all have the same area.

where n is the total number of players and the sum extends over all subsets S of N not containing player i. The formula can be interpreted as follows.

In his De dimensione parabolae of 1644, Toricelli considered a decreasing sequence of positive terms and showed the corresponding telescoping series necessarily converges to, where L is the limit of the sequence, and in this way gives a proof of the formula for the sum of a geometric series.

* Abel sum formula

D. J. Taylor argues that these factors, and the fact that Gollancz was not a person to part with such a sum on speculation, suggest that Gorer was confusing Orwell's eventual earnings from the book with a small contribution for out of pocket expenses that Gollancz might have given him.

Integral gender complementarity argues that men and women are each integral, whole beings unto themselves whose result when put together is greater than the sum of their parts.

He argues that because people seem to perceive not the sum of an experience but its average, it may be an instance of the representativeness heuristic.

At this point Einstein brings into play the first screen as well and argues as follows: since the incident particles have velocities ( practically ) perpendicular to the screen S < sub > 1 </ sub >, and since it is only the interaction with this screen that can cause a deflection from the original direction of propagation, by the law of conservation of impulse which implies that the sum of the impulses of two systems which interact is conserved, if the incident particle is deviated toward the top, the screen will recoil toward the bottom and vice-versa.

Integral complementarity differs from fractional complementarity, in that that it argues that men and women are each whole persons in and of themselves, and, together, equal more than the sum of their parts.

" In sum, Plato argues that philosophical pleasure is the only true pleasure since other pleasures experienced by others are simply a neutral state free of pain.

Frank argues that the race for status is bad for society as a whole, as there cannot be improvement in overall status available, because every time person A rises above person B, the sum of their status remains the same.

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

where the first sum is over positive energy states and the second over those of negative energy.

where p is the frequency of one allele and q is the frequency of the alternative allele, which necessarily sum to unity.

This shows in particular that is a natural number for any natural numbers n and k. There are many other combinatorial interpretations of binomial coefficients ( counting problems for which the answer is given by a binomial coefficient expression ), for instance the number of words formed of n bits ( digits 0 or 1 ) whose sum is k is given by, while the number of ways to write where every a < sub > i </ sub > is a nonnegative integer is given by.

The binomial coefficients appear as the entries of Pascal's triangle where each entry is the sum of the two above it.

According to the theorem, it is possible to expand the power ( x + y )< sup > n </ sup > into a sum involving terms of the form ax < sup > b </ sup > y < sup > c </ sup >, where the exponents b and c are nonnegative integers with, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.

This is particularly so in Sydney and Melbourne, where headways on many lines in the core of the network reach 3 – 5 minutes in peaks and 10 – 20 minutes off peak ( about 18 hours a day ) and enter an underground loop for passenger distribution in the city centre ; and where ridership per capita exceeds the sum of metro and commuter rail in comparable North American urban areas such as Toronto, Boston or the San Francisco Bay Area.

Contrast that to a P2P network, where its aggregated bandwidth actually increases as nodes are added, since the P2P network's overall bandwidth can be roughly computed as the sum of the bandwidths of every node in that network.

where is the vector sum of the physical forces applied to the particle and is the absolute acceleration ( that is, acceleration in a stationary frame ) of the particle, given by:

In fact, the sum of the magnetic pressure and plasma pressure must be constant everywhere on the heliosphere in order to have an equilibrium configuration: where the magnetic field is higher, the plasma must be cooler or less dense.

The size of a grammar is the sum of the sizes of its production rules, where the size of a rule is one plus the length of its right-hand side.

The common notation for the divergence ∇· F is a convenient mnemonic, where the dot denotes an operation reminiscent of the dot product: take the components of ∇ ( see del ), apply them to the components of F, and sum the results.

Equivalently, the determinant can be expressed as a sum of products of entries of the matrix where each product has n terms and the coefficient of each product is − 1 or 1 or 0 according to a given rule: it is a polynomial expression of the matrix entries.

Some ecological principles, however, do exhibit collective properties where the sum of the components explain the properties of the whole, such as birth rates of a population being equal to the sum of individual births over a designated time frame.

where D / Dt is the substantial derivative, which is the sum of local and convective derivatives.

The Dyson series can be alternately rewritten as a sum over Feynman diagrams, where at each interaction vertex both the energy and momentum are conserved, but where the length of the energy momentum four vector is not equal to the mass.

:, where is the sum over any n-sequence of length N.

: where is the sum over any k-sequence of length N.

Coulomb observed that, at failure, a distinct slip plane would form behind a sliding retaining wall and he suggested that the maximum shear stress on the slip plane, for design purposes, was the sum of the soil cohesion,, and friction, where is the normal stress on the slip plane and is the friction angle of the soil.

is an entire function in the variable b for all positive integers a where c < sub > d </ sub >( k ) is Ramanujan's sum.