“Sum”的概念、定义、习题解题方法及翻译-数学辞典

单词

Sum

释义

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 summationoperation can also be indicated using a capital sigma with upper and lower limits written above and below, and the indexindicated 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, each1 unit across and units high, where , 2, ..., . Now add a rotated copy on top, as in the above figure. Notethat the resulting figure has Width and Height , and so has Area . Thedesired sum is half this, so the Area of the boxes in the sum is . Since the boxes are of unit width, thisis 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 allterms 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, asstated 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

(33)

Other analytic sums include

(34)

(35)



			(36)

(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)

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'sStaircase, Gaussian Sum, Geometric Series, Gosper's Method, Hurwitz Zeta Function, InfiniteProduct, 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.

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