 # Sums

## Sums Topics

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### Knuth's series

Knuth's series is given by(1)(2)(3)(OEIS A096616), where is the Riemann zeta function (Knuth 2000; Borwein et al. 2004, pp. 15-17).

### Kempner series

sumOEIS023.10344A082839116.17696A082830219.25735A082831320.56987A082832421.32746A082833521.83460A082834622.20559A082835722.49347A082836822.72636A082837922.92067A082838A Kempner series is a series obtained by removing all terms containing a single digit from the harmonic series. Surprisingly, while the harmonic series diverges, all 10 Kempner series converge. For example,While they are difficult to calculate, the above table summarizes their approximate values as computed by Baillie (1979; Havil 2003, pp. 33-34).Schmelzer and Baillie (2008) have devised an improved algorithm for summing more general Kempner series, such as the sum of where the digits of contain no string 314. This sum has approximate value . In general, the when a particular string of length is excluded from the 's summed over is approximately given by (Baillie and Schmelzer 2008)...

### Ramanujan cos/cosh identity

The amazing identityfor all , where is the gamma function. Equating coefficients of , , and gives some amazing identities for the hyperbolic secant.

### Faulhaber's formula

In a 1631 edition of Academiae Algebrae, J. Faulhaber published the general formula for the power sum of the first positive integers,(1)(2)where is a generalized harmonic number, is the Kronecker delta, is a binomial coefficient, and is the th Bernoulli number.Computing the sums for , ..., 10 gives(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)

### Power sum

There are two kinds of power sums commonly considered. The first is the sum of th powers of a set of variables ,(1)and the second is the special case , i.e.,(2)General power sums arise commonly in statistics. For example, k-statistics are most commonly defined in terms of power sums. Power sums are related to symmetric polynomials by the Newton-Girard formulas.The sum of times the th power of is given analytically by(3)Other analytic sums include(4)(5)(6)for , where is a Pochhammer symbol. The finite version has the elegant closed form(7)for and 2. An additional sum is given by(8)An analytic solution for a sum of powersof integers is(9)(10)(11)where is the Riemann zeta function, is the Hurwitz zeta function, and is a generalized harmonic number. For the special case of a positive integer, Faulhaber's formula gives the sum explicitly as(12)where is the Kronecker delta, is a binomial coefficient, and is a Bernoulli number. It is also true that..

### Factorial sums

The sum-of-factorial powers function is defined by(1)For ,(2)(3)(4)where is the exponential integral, (OEIS A091725), is the En-function, is the real part of , and i is the imaginary number. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (OEIS A007489). cannot be written as a hypergeometric term plus a constant (Petkovšek et al. 1996). The only prime of this form is , since(5)(6)(7)is always a multiple of 3 for .In fact, is divisible by 3 for and , 5, 7, ... (since the Cunningham number given by the sum of the first two terms is always divisible by 3--as are all factorial powers in subsequent terms ) and so contains no primes, meaning sequences with even are the only prime contenders.The sum(8)does not appear to have a simple closed form, but its values for , 2, ... are 1, 5, 41, 617, 15017, 533417, 25935017, ... (OEIS A104344). It is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841,..

### Cumulative sum

A cumulative sum is a sequence of partial sums of a given sequence. For example, the cumulative sums of the sequence , are , , , .... Cumulative sums are implemented as Accumulate[list].

### Sum

A sum is the result of an addition. For example, adding1, 2, 3, and 4 gives the sum 10, written(1)The numbers being summed are called addends, or sometimes summands. The summation operation can also be indicated using a capital sigma with upper and lower limits written above and below, and the index indicated below. For example, the above sum could be written(2)The sum of a list of numbers is implemented as Total[list].A sum(3)in which each term is given by some fixed rule (i.e., is a well-defined sequence) is called a (finite) series, and if the number of terms is infinite, the sum is called an infinite series (or often just a "series"). A sum of the form(4)is called a geometric series.Conditions for convergence of a series can be determined in the Wolfram Language using SumConvergence[a, n].The general finite power sum(5)can be given by the expression(6)which is equivalent to Faulhaber's formula, where the notation means the quantity..

### Odd number theorem

The sum of the first odd numbers is a square number,A sort of converse also exists, namely the difference of the th and st square numbers is the th odd number, which follows from

### Nonnegative partial sum

Consider the number of sequences that can be formed from permutations of a set of elements such that each partial sum is nonnegative. The number of sequences with nonnegative partial sums which can be formed from the permutations of 1s and s (Bailey 1996, Brualdi 1997) is given by the Catalan numbers . For example, the permutations of having nonnegative partial sums are , , , , and (1, , 1, , 1, ).Similarly, the number of nonnegative partial sums of 1s and s (Bailey 1996) is given bywhere these coefficients form Catalan's triangle(OEIS A009766) and

### Nicomachus's theorem

The th cubic number is a sum of consecutive odd numbers, for example(1)(2)(3)(4)etc. This identity follows from(5)It also follows from this fact that(6)

### Alternating factorial

The alternating factorial is defined as the sum of consecutive factorialswith alternating signs,(1)They can be given in closed form as(2)where is the exponential integral, is the En-function, and is the gamma function.The alternating factorial will is implemented in the WolframLanguage as AlternatingFactorial[n].A simple recurrence equation for is given by(3)where .For , 2, ..., the first few values are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (OEIS A005165).The first few values for which are (probable) primes are , 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961, ... (OEIS A001272; extending Guy 1994, p. 100). Živković (1999) has shown that the number of such primes is finite. was verified to be prime in Jul. 2000 by team of G. La Barbera and others using the Certifix program developed by Marcel Martin.The following table summarizes the largest..

### Sophomore's dream

Borwein et al. (2004, pp. 4 and 44) term the expression of the integrals(1)(2)(3)(4)(OEIS A083648 and A073009)in terms of infinite sums "a sophomore's dream."For , write(5)(6)Integrating term by term then gives(7)(8)(9)(Borwein et al. 2004, p. 44).For , write(10)(11)Integrating term by term then gives(12)(13)(14)(Borwein et al. 2004, pp. 4 and 44).

### Lower sum

For a given function over a partition of a given interval, the lower sum is the sum of box areas using the infimum of the function in each subinterval .