## Sum

A sum is the result of an Addition. For example, adding 1, 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)

A simple graphical proof of the sum can also be given. Construct a sequence of stacks of boxes, each 1 unit across and units high, where , 2, ..., . Now add a rotated copy on top, as in the above figure. Note that the resulting figure has Width and Height , and so has Area . The desired sum is half this, so the Area of the boxes in the sum is . Since the boxes are of unit width, this is also the value of the sum.

The sum can also be computed using the first Euler-Maclaurin Integration Formula

 (3)

with . Then
 (4)

The general finite sum of integral Powers can be given by the expression

 (5)

where the Notation means the quantity in question is raised to the appropriate Power and all terms of the form are replaced with the corresponding Bernoulli Numbers . It is also true that the Coefficients of the terms in such an expansion sum to 1, as stated by Bernoulli without proof (Boyer 1943).

An analytic solution for a sum of Powers of integers is

 (6)

where is the Riemann Zeta Function and is the Hurwitz Zeta Function. For the special case of a Positive integer, Faulhaber's Formula gives the Sum explicitly as
 (7)

where is the Kronecker Delta, is a Binomial Coefficient, and is a Bernoulli Number. Written explicitly in terms of a sum of Powers,
 (8)

Computing the sums for , ..., 10 gives

 (9) (10) (11) (12) (13) (14) (15) (16) (17) (18)

Factoring the above equations results in

 (19) (20) (21) (22) (23) (24) (25) (26) (27) (28)
From the above, note the interesting identity

 (29)

Sums of the following type can also be done analytically.

 (30) (31) (32)

By Induction, the sum for an arbitrary Power is
 (33)

Other analytic sums include

 (34)

 (35)

 (36)

so
 (37)

 (38)

 (39)

 (40)

 (41)

To minimize the sum of a set of squares of numbers about a given number

 (42)

take the Derivative.
 (43)

Solving for gives
 (44)

so is maximized when is set to the Mean.

See also Arithmetic Series, Bernoulli Number, Clark's Triangle, Convergence Improvement, Dedekind Sum, Double Sum, Euler Sum, Factorial Sum, Faulhaber's Formula, Gabriel's Staircase, Gaussian Sum, Geometric Series, Gosper's Method, Hurwitz Zeta Function, Infinite Product, Kloosterman's Sum, Legendre Sum, Lerch Transcendent, Pascal's Triangle, Product, Ramanujan's Sum, Riemann Zeta Function, Whitney Sum

References

Boyer, C. B. Pascal's Formula for the Sums of the Powers of the Integers.'' Scripta Math. 9, 237-244, 1943.

Courant, R. and Robbins, H. The Sum of the First Squares.'' §1.4 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 14-15, 1996.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, 1996.