sums of powers of integers

Sums of Powers of IntegersAuthor(s): B. L. Burrows and R. F. TalbotSource: The American Mathematical Monthly, Vol. 91, No. 7 (Aug. - Sep., 1984), pp. 394-403Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2322985 .

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SUMS OF POWERS OF INTEGERS

B. L. BURROWS AND R. F. TALBOT Department of Mathematics, North Staffordshire Polytechnic, Beaconside, Stafford, England ST] 8 OA D

1. Introduction. The Bernoulli family produced several mathematicians, the first of whom was Jacob (1654-1705). He was the son of Nicolaus, a merchant in Basle, and it is reported [1] that Jacob had the motto 'Invito patre sidera verso' (against my father's will I study the stars). Despite his father's opposition, Jacob devoted his life to the study of mathematics and astronomy. His most famous work is 'Ars Conjectandi,' which was published after his death, in 1713, and which reflected his interest in probability. It was in this book that the famous Bernoulli numbers were introduced in connection with finding sums of powers of integers. Jacob was very enthusiastic about the technique that he had developed and in comparing his calculations with those of Bullialdus (1605-1694) he states [2] 'with the help of this table it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum

91,409,924,241,424,243,424,241,924,242,500.' In this article we show that a simple formula will give highly accurate answers to such sums and with the help of a very crude electronic calculator we can obtain, in 9 seconds,

1000

L r10 9.14104 X 1031 r=1

which is in error by 0.0005%! The approximation used was

(1.1) E rk' (n?2)k+1

r=1 and the idea stems from another observation of Bernoulli's that the sums can be expressed in the form p([n(n + 1)]) when k is odd and (2n + 1)p([n(n + 1)]) when k is even, where p(x) is a polynomial [3]. We refine the approximation (1.1) and extend the results to the case where k is any real number.

2. E Ir k (k is a Positive Integer). It is well known that this sum can be expressed as a polynomial in n of degree k + 1. Bernoulli's observation mentioned earlier suggests that there is some advantage to be gained by considering the equivalent polynomial in n + 4 which is also of degree k + 1.

We shall use the method of undetermined coefficients to find this polynomial which we shall denote by Sk+l(n + 4). Thus

n (2.1) E rk= Sk+ n + -2-

r==12

Brian L. Burrows: I obtained my B.Sc. from the University of Surrey in 1968 and my doctorate from Nottingham University in 1971 for research into quantum mechanics. Since then I have published work in numerical analysis, variational methods, combinatorics and quantum theory. My nonmathematical interests include campanol- ogy, avoiding administration, and being beaten at squash by Dr. Talbot.

Richard F. Talbot: I was awarded my B.Sc. by Nottingham University in 1966 and I obtained my doctorate from the same university in 1969 for research in the area of fluid mechanics. Since that time my mathematical interests have diversified into combinatorics and more recently numerical analysis. Apart from my family my main nonmathematical interests are local history, sport in general including cricket (similar to baseball but more intellectual!), and squash.

394



SUMS OF POWERS OF INTEGERS 395

where / 1\ ~k+1

(2.2) Sk+i(ln+ 2= Eat(n +

From (2.1) it follows that n-I/

(2.3) E = Sk+( n

and so equations (2.1) and (2.3) give

(2.4) Sk+1( n + )Sk+1 n -)- nk.

Now, using (k) to denote the familiar binomial coefficient we have

(2.5) n( 2) ( _ 2) ,E (y n jl+-)

so that the right-hand side is either an even or an odd polynomial in n. Equation (2.4) can be rewritten

k-i1 t

(2.6) F, at (J)(2 n t

(I +( 1+) = n k

t=i =

Notice that a0 will have to be determined independently. The coefficients At for t = k + 1 k, . . . , can now be found by equating in (2.6) the terms in nP for p = k, k - 1, . . 0. This gives the following results:

ak+ + 1) 1 (2.7) ak+(1 = 1,

a(k) = 0,

(2.7) ak*1( (2)+a- 1) ?

ak(3 )( )+ ak2( ) = 0,

and generally

ak+(k + 1)(1 )P+(1 +(-1)p+2) + ak (k)(2) (1 +(_i)P+1)

(a- k - ( P-1 (I (- 1 )) + *+ a k+ 1 ( k + 1-p =)0

We first observe that the second of these results immediately gives ak = 0 and hence ak-2, ak-4, ... are all zero. From the first of these equations we obtain ak+l and then solving for akl,ak-3,... in turn, we have

1 ak+i = k + 1' ak+l =-

( (k (2.8) ak-3=ona



396 B. L. BURROWS AND R. F. TALBOT [August-September

ak-5=-31 (k) k=2~ 7-63 kY

127 k ak7 2 7*240 \7

Thus, for example, when k = 3 we obtain

L r3 = s4=n + 2 n + - + n + ao. r=1 2 S( 4) 2( 8[

When n = 1, S4(3) = 1 and so ao = 1 Therefore,

/ 1\ 1/I 1\~2 2

(2.9) S4( + ) 4{(n + -

= -n2(n + 1)2, 4 this final form being the one that is well known.

The results above can perhaps be better appreciated from a computational point of view. Writing X for N + 2 and S(k) for y.r k, then we have shown that

(2.10) S(k) = ak+lXk+l + ak-lXkl1 + ak-3Xk-3 +

where the coefficients are given by (2.8) and the last term is ao if k is odd and a, X if k is even. Thus when k > 4, we may rewrite (2.10) in the form

(2.11) S(k) = k+ 1 (1_(2 )+ (__ 4)_ k + 1 12 X2 240 X4

From (2.11) it is clear that good results may be obtained from the approximation

k + i (2.12) S(k) Xk +1.

We may illustrate these results by considering E35= r5 and, comparing the exact value of 333,263,700 with just two approximations that can be obtained from (2.10) or equivalently (2.11).

The first approximation, expressed in (2.12), gives

L r5 _3__ = 333,594,489,

r=1l

which is in error by about 0.1%. The second approximation, using just the first two terms of (2.10) or (2.11) gives

r5 6 - 35 54 = 333,263,608

which is in error by 92 or about 0.0000003%. The natural extension of this work is to the case where k is not necessarily an integer.

3. The General Case. In this section we generalize the previous result to d-cal with EZ=1r k for all real k deriving a result that extends (2.11). In particular we show that the simple approximation obtained in (2.12) provides surprisingly accurate estimates when k is positive.

The formulae we will derive for k not a positive integer result from an application of the methods of asymptotic analysis to sums of powers of integers. A discussion of these techniques appears in [4]; in particular equation 3.4.6 of [4] gives a result for the convergent case in terms of



1984] SUMS OF POWERS OF INTEGERS 397

powers of n and the Riemann Zeta function. Consider the Taylor series expansion of f(x) = xk about the integer x = r.

(3.1) f(x) = f(r) +(x - r)f'(r) + (-2! f"(r) +

+ (x _r) fq( ) + (x r 1(Zr)'

where Zr = r + Or(x - r) and 0 < Or < 1. Using the notation

(3.2) (k) k(k- l) (k- r+ 1)

for non-integer k and integer r, we may rewrite (3.1) in the following way, where we have chosen q to be even without loss of generality.

(3.3) xk = rk +( )rk1(X - r) +(2)rk2(X - r)2 +

1 +( kq

r k-q(X - r) q +

k zrk-q l(X-_r) q1

Integrating from r - 2 to r + 2 then gives

(3.) fr (i

kdX k(k)kr (1)3(k)k4r2(1)5 (3.4) xkdx 1/2+ 2 ) k 3 (2 ) k4 5 (2)+

Summing equation (3.4) from r = 1 to n yields

(3.5) g(k) = r+ k 2 ) Er + ( 4 E r

k +

where

g(k) =k 1(n + 2) -2 (k 0 - 1)

and g(-1) = log(2n + 1). Write S(k) = En.4rk. Then equation (3.5) may now be written

(3.6) g(k)=S(k) + k S(k- 2)+ k

S(k- 4)+

+( k(q I 1)2qS(k - q) + Rq+

where

R ( ~k ) r+ (1/2) k--( )q+ldX Rq+1 =q f+ )f?(1/2)Zir1(x -r)?d

and Zr iS given in (3.1). Thus for q > k - 1,

k n r+ p 1\2) k-q-1

JRq+1I < (qf ,i r -~ - jx - rjq?1 dx, q + I r=1 - (1/2) 2,

that is,

(3.7) (q< k - r 1 2() IR+i q + I)r=1( 2 q +2




and so

jRq+ij <

From this we can easily deduce that for fixed k and n, lirn q4 R q + 1=0 so that the series (3.6) converges.

Putting k, k - 2, k - 4,... in turn in equation (3.6) gives the set of equations (3.8)

g(k) = S(k)?( k S(k -2) +(k)IS(k -4) +(k SI 6 \213.-2 2415 .2 4 \617 26(

g(k -2) = S(k -2) + ( )S(k -4)?(k2) 1S(k -6) +.. k213 .22241 5 24

g(k -4) = S(k -4) +(k4)l2 S(k -6) s..

g(k -6) = S(k -6) +

These equations can be written in matrix form

(3.9) g =MS,

where

g(k) 1 (~3X-22 (~5-24 (6I7-2 6

g(k -2) 0 1 (k -2) g= g(k-4) ,M= 0 0 1 2 (k4 )l2

0 0 0 1..

and

We initially solve the problem by truncating the infinite matrix M to a finite size Ml with 1 rows and columns. In a similar way the vector g is truncated to g1 and s1 is defined so that M1s1 = g1.

It is worth pointing out that the form of the matrix M is such that having found M7-1 for a given I then this M71' forms a submatrix of the inverses of higher orders. Taking as a particular case 1 = 4, then we obtain

1 2( 3 1 22 (4)1i2 4 3 126

(3.10) 0 k2)3l (k4 52)7

0 0 1 (k- 4)

0 0 0 1




so that

(3.11) S4 = MV9g4.

From this, we can obtain a 4-term approximation to S(k) given by

(3.12) S4(k) = g(k) -() 12g(k - 2) + () 7 4g(k - 4) -() 31 6g(k - 6).

Equation (3.12) agrees with the previous results (equations (2.8) and (2.10)) obtained when k was a positive integer.

Again the numerical accuracy of the result may be demonstrated by taking the first few terms as in the next example.

Now, E35 jr25 = 76138.722369 and the table below gives the successive approximations for various values of 1.

1 Approximation 1 76160.721097 2 76138.725044 3 76138.722204 4 76138.722459

Note that the approximation 35.535/3.5 which is analogous to (2.12) is in error by only .003%. The matrix technique used in this section can be applied in the case where k is a positive

integer. M will now be of finite order m and the previous analysis will still apply provided that the

restricted order 1 is less than m.

4. Modifications when k is Negative. In practice, when k is negative, the accuracy of (3.12) is improved by starting the summation at r = p. This is equivalent to redefining S(k) by

n

(4.1) S(k) = E rk, r=p

with the corresponding modifications

(4.2) g(k ((

+)+ - p k k+

and

(4.3) g(-1) = log(n + - log( P

For example, in the case k = -1, we have the explicit form

(4.4) S: - = log(n + - log(p - + n + V (p-

- 960 + 2) -(p - 2 + 632 n + 2) (p - )

Thus for p = 3 and n = 50, we have used the inverse matrix technique described previously to compute a 1-term approximation of 3.0056826 and a 5-term approximation of 2.9992059. This can be compared with the exact result of 2.9992053 to 7 decimal places.

It is also interesting to compare the well-known approximation n r

(4.5) E logn + -y, r=l




where y = 0.57721566... is Euler's constant, with our first approximation

(4.6) ~~~~~~n p- 1 n + 2 (4.6) r1 I + log

{ }

r r=r

and our second approximation

(4.7) 1 -+ {log + 2 (( 2 ( 2 r=1 r r=1 r1 2

For n = 50, equation (4.5) leads to the approximation 4.48924 compared with the exact value of 4.499205, an error of .22%. The following table compares this with the approximations (4.6) and (4.7) for various values of p.

Approximation p = 2 p = 3 p = 4 (4.6) 4.51651 4.40468 4.50254 (4.7) 4.49801 4.49903 4.49916

So far we have only considered examples of finite series of the form n= Irk since for k >? -1 the sum to infinity does not exist. However, equations (3.12) and (4.2) can be used to estimate ZO=Irk when k < -1. In this case we may let n -s o and obtain

00 Pl1k1 -

(4.8) E rk= E rk p + k )p _ r=1 r=1 k 1 22 1 k 1 2

4 240 k - 3 -2 ) +(6) 1344 k - 5 (-2) -

Equation (4.8) can be written 00 P-1 1 /l\k+I

k, ik-1 i 1k-3

(4.9) rk r r~- I - + - -( - -I'-i' )rEik rE1r k + 1 (P 2) 24(P 2 3960 2

5 8064 ( 2 - 2) -

We can easily adapt the proof of convergence of (3.6) to the case where n is infinite (and of course k < - 1). From a modified form of (3.7) we have

lRqI < (k (r -

2q 1) q 2r=p /

Since k < - 1, the infinite sum is convergent so that

lim Rq= 0. q -o0

Computations using the terms given in (4.9) have been carried out for the infinite series where k = -2 and k = - 3.5 with various values of p. The results are shown below.

k p = 2 p = 3 p = 4 Exact - 2 1.644466 1.644928 1.644934 1.644934 - 3.5 1.124871 1.126723 1.126733 1.126730




5. Convergence. In the previous sections we have approximated En rk by taking the first few terms of the series

00

(5.1) E m1g(k - 21), 1=0

where ml is the Ith entry in the first row of Ml71 (see (3.9)-(3.12)) and g(k) is given by (4.2). In fact (5.1) converges only when k is a positive integer and in this case the series breaks off

after finitely many terms. In all other cases the series (5.1) diverges and we now show that its terms eventually have

increasing modulus. We first note that from (4.2)

g(k - 21) = - (n + 1 )k + ( - )2) )}

so that for sufficiently large,

(5.2) Ig(k - 21)1 >1k - 21+ 11 21k1 forp ?1.

The matrix Ml can be written in the form (5.3)

2 1-2 1 a(k) X1a(k)a(k - 2) X2 17 a(k - 2i) ... X1-2 T a(k - 2i)

1- 2 0 1 a(k-2) X1a(k-2)a(k-4) ... X-3 H a(k-2i)

Ml ~~~~~~~~~~~~~~~~~~1- 2 0 0 1 a(k-4) ... Xl14 Y a(k-2i)

a(k - 21 + 4) 0 0 0 0 1

where

a(k) =(213 222 and Xr (2r+ 3)!

Consider the equation (5.4) Mix= e, where e is the Ith column of the I x I identity matrix. The first entry of x is xl = ml. Using Cramer's Rule and noting that det Ml = 1, we have (5.5)

2 1-2 0 a(k) Xla(k)a(k - 2) 2 171a(k - 2i) ... X2 U a(k - 2i)

1- 2

0 1 a(k-2) X1a(k-2)a(k-4) ..

X3Yrl1a(k-22i)

0 0 a(k- 21+ 4) 1 0 1




2 1-2 a(k) Xla(k)a(k - 2) X2 H a(k - 2i) ... X-2 El a(k - 2i) i=O i=0

1 a(k-2) Xla(k-2)a(k - 4) ... 0 1 a(k-4) ...

0 0 0 a(k-21+4)

Use induction on I to prove that

(5.6) m= (-1)'+la(k)a(k - 2) * a(k - 21 + 4)A,

where

Ix 2 ... 1I-2

1 1 xi1 ... X1-3 0 1 1

(5.7) AI= 0 : ; 1 X1 0 0 ... 1

The value of A, can be found by successively eliminating the 'ones' below the main diagonal by subtracting a multiple of the previous row. This produces a triangular determinant given by

1 X1 X -2 0 f,

(5.8) A,= 0 f2

0 0 ... f1-2

whose value is f112 1.. f1-2- Increasing / does not change the previous values of f, so that for example

(5 -9) Al+ I = flf2 .. fl t-2 fl-l1-

A numerical calculation of the sequence { fi } shows that we have a monotonic decreasing sequence tending rapidly to a limit. The table below gives various values of fi.

i fi 1 0.7 2 0.63265306

13 0.60792713 14 0.60792711 15 0.60792710 16 0.60792710

Defining this limit to be 1/3, we see that A, 0 as I oo. However, from the monoticity we can conclude that

(5.10) A (A)




Now H-I-2a(k - 2i) can be written as

(5.11) k(k-1) ... (k-21+ 3) 241-1

Thus combining the results of this section, we have shown that the general term of (5.1) satisfies

(5.12) m,g(k - 21)j >k(k - 1) ... (k - 2/ + 3)1 p k- 21+ 1

(5.12) lmlg(k - 21)1 > (24/3)1-1 k - 21 + 1 Since p and k are fixed, (5.11) shows that eventually the terms of the series (5.1) increase in

magnitude. This shows that these approximations will always get worse for sufficiently large 1. The reason why the estimates given in this paper are good can be seen by regarding equations (3.8) truncated in the form

(5.13) MIS/ = g, + ?1, where SI is the first I values of the exact solution and - e1 is the vector of the truncation errors in the first I equations (see (3.7)). The exact expression for S(k) is then a finite sum of terms in the form

(5.14) m,(g(k - 2t) + t),g where m, are the elements of e/. Taking the particular case t = 3, we have

(5.15) = 7k(k - 1)(k - 2)(k - 3) M3 ~4! .15 -24

Since % is small and for positive k the sums are of the order (n + k)k+1/(k + 1), then for large n the relative error

(5.16) Im3A3I/S( k) is small. For negative k (particularly k < -1) the approximations are formed by using p > 1. Since X93 decreases with p, then it is always possible to reduce the size of ImA331. These arguments hold for any t with sufficiently large n or p so that it is always possible to obtain good estimates from the initial terms of (5.1). This gives a heuristic argument why we have been able to obtain surprisingly good approximations to S(k) from a series of terms (5.1) that actually diverges!

References

1. E. T. Bell, Men of Mathematics, Penguin, New York, 1937. 2. D. E. Smith, A Source Book in Mathematics, McGraw-Hill, New York, 1929. 3. A. W. F. Edwards, Sums of powers of integers: a little of the history, Math. Gaz. (1982) 22-28. 4. N. G. de Brujn, Asymptotic Methods in Analysis, North-Holland, Amsterdam, 1958.

130. MISCELLANEA

These forms were like the cunning tables used by mathematicians, which may be entered from top, bottom, right, and left, which entrances consist of scores of lines and dozens of columns, and from which may be drawn, without reasoning or thinking, thousands of different conclusions, all unchallengeably precise and true.

-Martin Eden, by Jack London, Macmillan, New York, 1973, p. 226.



sums of powers of integers

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