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Power Sum

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There are two kinds of power sums commonly considered. The first is the sum of pth powers of a set of n variables x_k,

S_p(x_1,...,x_n)=sum_(k=1)^nx_k^p,
(1)

and the second is the special case x_k=k, i.e.,

S_p(n)=sum_(k=1)^nk^p.
(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 k times the kth power of x is given analytically by

sum_(k=0)^nkx^k=(x-(n+1)x^(n+1)+nx^(n+2))/((x-1)^2).
(3)

Other analytic sums include

(sum_(k=0)^(infty)x^k)^p=(1-x)^(-p)
(4)
=1/((p-1)!)sum_(n=0)^(infty)((n+p-1)!)/(n!)x^n
(5)
=1/((p-1)!)sum_(n=0)^(infty)(n+1)_(p-1)x^n
(6)

for |x|<1, where (n)_p is a Pochhammer symbol. The finite version has the elegant closed form

(sum_(k=0)^nx^k)^p=1/((p-1)!)sum_(k=0)^(np)((n-|n-k|+p-1)!)/((n-|n-k|)!)x^k
(7)

for p=1 and 2. An additional sum is given by

(sum_(n=0)^inftya_nx^n)^2=sum_(n=0)^inftya_n^2x^(2n)+2sum_(n=1; i+j=n; i<j)^inftya_ia_jx^n.
(8)

An analytic solution for a sum of powers of integers is

S_p(n)=sum_(k=1)^(n)k^p
(9)
=zeta(-p)-zeta(-p,1+n)
(10)
=H_n^((-p)),
(11)

where zeta(z) is the Riemann zeta function, zeta(z;a) is the Hurwitz zeta function, and H_n^((k)) is a generalized harmonic number. For the special case of p a positive integer, Faulhaber's formula gives the sum explicitly as

S_p(n)=1/(p+1)sum_(k=1)^(p+1)(-1)^(delta_(kp))(p+1; k)B_(p+1-k)n^k,
(12)

where delta_(kp) is the Kronecker delta, (n; k) is a binomial coefficient, and B_k is a Bernoulli number. It is also true that the coefficients of the terms in such an expansion sum to 1, as stated by Bernoulli (Boyer 1943).

Bernoulli used the property of the figurate number triangle that

sum_(i=0)^na_(ij)=((n+1)a_(nj))/(j+1),
(13)

along with a form for a_(nj) which he derived inductively to compute the sums up to n=10 (Boyer 1968, p. 85). For p in Z>0, the sum is given by

sum_(k=1)^nk^p=((B+n+1)^([p+1])-B^([p+1]))/(p+1),
(14)

where the notation B^([k]) means the quantity in question is raised to the appropriate power k, and all terms of the form B^m are replaced with the corresponding Bernoulli numbers B_m. Written explicitly in terms of a sum of powers,

sum_(k=1)^(n)k^p=n^p+sum_(k=0)^(p)(B_kp!)/(k!(p-k+1)!)n^(p-k+1)
(15)
=sum_(k=1)^(p+1)((-1)^(p-k+1)B_(p-k+1)p!)/(k!(p-k+1)!)n^k
(16)
=sum_(k=1)^(p+1)b_(pk)n^k,
(17)

where

b_(pk)=((-1)^(p-k+1)B_(p-k+1)p!)/(k!(p-k+1)!).
(18)

It is also true that the coefficients of the terms b_(pk) sum to 1,

sum_(k=1)^(p+1)b_(pk)=1,
(19)

which Bernoulli stated without proof.

A double series solution for S_p(n) is given by

S_p(n)=sum_(i=1)^psum_(j=0)^(i-1)(-1)^j(i-j)^p(n+p-i+1; n-i)(p+1; j).
(20)

Computing the sums for p=1, ..., 10 gives

sum_(k=1)^(n)k=1/2(n^2+n)
(21)
sum_(k=1)^(n)k^2=1/6(2n^3+3n^2+n)
(22)
sum_(k=1)^(n)k^3=1/4(n^4+2n^3+n^2)
(23)
sum_(k=1)^(n)k^4=1/(30)(6n^5+15n^4+10n^3-n)
(24)
sum_(k=1)^(n)k^5=1/(12)(2n^6+6n^5+5n^4-n^2)
(25)
sum_(k=1)^(n)k^6=1/(42)(6n^7+21n^6+21n^5-7n^3+n)
(26)
sum_(k=1)^(n)k^7=1/(24)(3n^8+12n^7+14n^6-7n^4+2n^2)
(27)
sum_(k=1)^(n)k^8=1/(90)(10n^9+45n^8+60n^7-42n^5+20n^3-3n)
(28)
sum_(k=1)^(n)k^9=1/(20)(2n^(10)+10n^9+15n^8-14n^6+10n^4-3n^2)
(29)
sum_(k=1)^(n)k^(10)=1/(66)(6n^(11)+33n^(10)+55n^9-66n^7+66n^5-33n^3+5n)
(30)

or in factored form,

sum_(k=1)^(n)k=1/2n(n+1)
(31)
sum_(k=1)^(n)k^2=1/6n(n+1)(2n+1)
(32)
sum_(k=1)^(n)k^3=1/4n^2(n+1)^2
(33)
sum_(k=1)^(n)k^4=1/(30)n(n+1)(2n+1)(3n^2+3n-1)
(34)
sum_(k=1)^(n)k^5=1/(12)n^2(n+1)^2(2n^2+2n-1)
(35)
sum_(k=1)^(n)k^6=1/(42)n(n+1)(2n+1)(3n^4+6n^3-3n+1)
(36)
sum_(k=1)^(n)k^7=1/(24)n^2(n+1)^2(3n^4+6n^3-n^2-4n+2)
(37)
sum_(k=1)^(n)k^8=1/(90)n(n+1)(2n+1)(5n^6+15n^5+5n^4-15n^3-n^2+9n-3)
(38)
sum_(k=1)^(n)k^9=1/(20)n^2(n+1)^2(n^2+n-1)(2n^4+4n^3-n^2-3n+3)
(39)
sum_(k=1)^(n)k^(10)=1/(66)n(n+1)(2n+1)(n^2+n-1)(3n^6+9n^5+2n^4-11n^3+3n^2+10n-5)
(40)

(OEIS A064538 and A079618).

Sumk

A simple graphical proof of the special case of S_1(n)=n(n+1)/2 can also be given by constructing a sequence of stacks of boxes, each 1 unit across and k units high, where k=1, 2, ..., n. Now add a rotated copy on top, as in the above figure. Note that the resulting figure has width n and height n+1, and so has area n(n+1). The desired sum is half this, so the area of the boxes in the sum is n(n+1)/2. Since the boxes are of unit width, this is also the value of the sum.

The sum S_1(n)=n(n+1)/2 can also be computed using the first Euler-Maclaurin integration formula

sum_(k=1)^nf(k)=int_1^nf(x)dx+1/2f(1)+1/2f(n)+1/(2!)B_2[f^'(n)-f^'(1)]+...
(41)

with f(k)=k. Then

sum_(k=1)^(n)k=int_1^nxdx+1/2·1+1/2·n+1/6(1-1)+...
(42)
=1/2(n^2-1)-1/2+h+1/2n
(43)
=1/2n(n+1).
(44)
CubeSquare

The surprising identity

S_3(n)=sum_(k=1)^(n)k^3
(45)
=(sum_(k=1)^(n)k)^2,
(46)

known as Nicomachus's theorem, can also be illustrated graphically (Wells 1991, pp. 198-199).

Schultz (1980) showed that the sum S_p(n) can be found by writing

S_p(n)=c_(p+1)n^(p+1)+...+c_1n
(47)

and solving the system of p+1 equations

sum_(i=j+1)^(p+1)(-1)^(i-j+1)(i; j)c_i=delta_(j,p)
(48)

obtained for j=0, 1, ..., p (Guo and Qi 1999), where delta_(j,p) is the Kronecker delta. For example, the three equations to be solved for p=2 are

0=c_1-c_2+c_3
(49)
0=2c_2-3c_3
(50)
1=3c_3,
(51)

giving c_1=1/6, c_2=1/2, and c_3=1/3, or

S_2(n)=1/6n+1/2n^2+1/3n^3,
(52)

as expected.

S_i(n) is related to the binomial theorem by

(1+n)^(k+1)=1+sum_(i=0)^k(k+1; i)S_i(n)
(53)

(Guo and Qi 1999).

See also

Diophantine Equation, Faulhaber's Formula, Multigrade Equation, Nicomachus's Theorem, Series, Sum

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References

Boyer, C. B. "Pascal's Formula for the Sums of Powers of the Integers." Scripta Math. 9, 237-244, 1943.Boyer, C. B. A History of Mathematics. New York: Wiley, 1968.Brualdi, R. A. Introductory Combinatorics, 3rd ed. New York: Elsevier, p. 119, 1997.Cao, J.-T. "A Method of Summing Series and Some Corollaries" [Chinese]. Math. Pract. Th. 20, 77-84, 1990.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 106, 1996.Guo, S.-L. and Qi, F. "Recursion Formulae for sum_(m=1)^(n)m^k." Z. Anal. Anwendungen 18, 1123-1130, 1999.Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 81-82, 2003.Schultz, H. J. "The Sums of the kth Powers of the First n Integers." Amer. Math. Monthly 87, 478-481, 1980.Sloane, N. J. A. Sequences A064538 and A079618 in "The On-Line Encyclopedia of Integer Sequences."Struik, D. A Source Book in Mathematics, 1200-1800. Cambridge, MA: Harvard University Press, 1969.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 198-199, 1991.Yang, B.-C. "Formulae Related to Bernoulli Number and for Sums of the Same Power of Natural Numbers" [Chinese]. Math. Pract. Th. 24, 52-56 and 74, 1994.Zhang, N.-Y. "Euler's Number and Some Sums Related to Zeta Function" [Chinese]. Math. Pract. Th. 20, 62-70, 1990.

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Power Sum

Cite this as:

Weisstein, Eric W. "Power Sum." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PowerSum.html

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