Sums of powers of the natural numbers

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1 Sums of powers of the atural umbers Dr Richard Kederdie Kederdie Maths Tutorig December 04 A commo questio asked of studets of mathematical iductio is to prove = ( + ) ( + ). This ote derives the expressio o the right-had side as 6 well as expressios for higher powers. Sum of the first atural umbers The series is a Arithmetic Progressio with first term =, differece = ad terms. The stadard formula for such a series is (a + l) = ( + ) which we ca also write as +. Thus we have The telescopic property The telescopic property is useful for derivig results relatig to series. Cosider k = ( + ) = + () [a(k + ) - a(k)] = [a() - a()] + [a(3) - a()] + [a(4) - a(3)] +...[a() - a( - )] + [a( + ) - a()] () = a( + ) - a() (3) Sum of the first square umbers Cosider Usig the telescopic property this becomes (k + ) 3 - k 3 (4) (k + ) 3 - k 3 = ( + ) 3-3 (5) = (6) Alteratively we ca expad out the summatio (k + ) 3 - k 3 = k k + 3 k + - k 3 (7) = 3 k + 3 k + (8) = 3 k + 3 ( + ) + (9) Equatig (6) with (0), = 3 k (0) 3 k = ()

2 Sums of Powers report.b So k = () = (3) or i factored form k = 6 ( + ) ( + ) (4) Sum of the first cubed umbers Cosider Usig the telescopic property this becomes (k + ) 4 - k 4 (5) (k + ) 4 - k 4 = ( + ) 4-4 (6) = (7) Alteratively we ca expad out the summatio (k + ) 4 - k 4 = k k k + 4 k + - k 4 (8) = 4 k k + 4 k + (9) = 4 k 3 + ( + ) ( + ) + 4 ( + ) + (0) Equatig (7) with (), So = 4 k () 4 k = () k 3 = (3) = (4) or i factored form k 3 = ( + ) (5) Note that sice (+) is the sum of the first atural umbers the we have = ( ) (6)

3 Sums of Powers report.b 3 Sum of the first quartic umbers Proceedig as previously which becomes (k + ) 5 - k 5 = ( + ) 5-5 (7) 5 k k k + 5 k + = (8) ad usig substitutios for k 3, k ad k from above (8) simplifies to or i factored form Summary so far The sums of the first four powers of the atural umbers are: k 4 = (9) k 4 = 30 ( + ) ( + )[3 ( + ) - ] (30) k = +. k = k 3 = k 4 = Cosiderig the geeral power p, the similarities of the expressios for k p iclude the first term is p d the secod coefficiet is powers of decrease from p+ to (the coefficiets for whe p = 3 ad whe p = 4 are 0) the coefficiets sum to i each case I factored form we have k = ( + ). k = ( + ) ( + ) 6 k 3 = [ ( + )] 4 k 4 = ( + ) ( + )[3 ( + ) - ] 30 It ca be see that the RHS is a polyomial i (+) with the extra factor + whe p is eve.

4 4 Sums of Powers report.b Historical ote Archimedes (87- BC) stated ad proved a formula for the sum of squares as Propositio 0 i his treatise traslated as O Spirals: If a series of ay umber of lies be give, which exceed oe aother by a equal amout; ad the differece be equal to the least, ad if other lies be give equal i umber to these ad i quatity to the greatest, the squares o the lies equal to the greatest, plus the square o the greatest ad the rectagle cotaied by the least ad the sum of all those exceedig oe aother by a equal amout will be the triplicate of all the squares o the lies exceedig oe aother by a equal amout. I moder terms, Archimedes is referrig to a arithmetic progressio with differece equal to the first term. If we make both equal to the the propositio states: ( + ) + ( ) = (3) This is illustrated below for = 4 where there are 5 lots of 4 plus ( ) cut up ito 3 lots each of,, 3 ad 4 : Now usig ( ) = ( + ) we have ( + ) + ( + ) = (3) as previously derived. ( + ) + = (33) ( + ) ( + ) = (34) 6 ( + ) ( + ) = (35) Besides the aciet Greeks, Hidus ad Arabs had rules for summig powers up to ad icludig 4. I 63 Joha Faulhaber ( ) published Academiae Algebrae that cotaied the sums up to p = 7. However there was o geeral formula for arbitrary p util Jacques (some refereces use Jakob or Jacob) Beroulli ( ) provided a solutio that was ot published util 73 i Ars Cojectadi. Beroulli listed the results up to power 0 ad described the geeral patter, without proof. The patter required the use of what came to be termed `Beroulli umbers` that have o obvious patter but ca be geerated i a umber of ways. Beroulli umbers Oe way of obtaiig these umbers is from the coefficiets i the power series expasio of x B x e x - = =0! (36) Mathematica gives the expasio, up to x, as

5 Sums of Powers report.b 5 x Series, {x, 0, } Exp[x] - - x + x which ca be writte as - x x x x x O[x]3 - x! + 6 x! - 30 x 4 4! + 4 x (37) 6! Thus the Beroulli umbers are (B 0 =, B = -, B = 6, B 3 = 0, B 4 = - 30, B 5 = 0, B 6 = 4, B 7 = 0,...). Note that B = 0 for odd except =. Aother way is the use of Beroulli polyomials, B (x), which appear i the expasio t e x t B (x) t e t - = =0! (38) Mathematica gives the expasio, up to t 6, as Series t ex t, {t, 0, 6} e t x t x + 6 x t + x - 3 x + x 3 t x - 60 x x 4 t x + 0 x3-5 x x 5 t x + 05 x 4-6 x x 6 t O[t] 7 This expasio ca be re-writte as t0 0! + x - t! + x - x + 6 t! + x - 3 x + x t3 3! + x4 - x 3 + x - 30 t (39) 4! The Beroulli polyomials are the coefficiets of t ad the Beroulli umbers are the costat terms of the polyomials. Oe property of the Beroulli umbers is k +! B k + k + The geeral formula usig Beroulli umbers I moder otatio we ca write B k- + k + 3 B k k + B k 0 = 0 (40) k = 0 + (4) k = B (4) Note that i (43) B 3 = 0. The patter ca be summarised as k 3 = B B 3 (43) k p = p + p + 0 p+ + p + p p + p + B j j p+- j (44) j=

6 6 Sums of Powers report.b Let s test this by summig powers of the first 0 atural umbers. First set up a fuctio for arbitrary ad p. I[4]:= sumpower[_, p_] := p + Biomial[p +, 0] p+ + Biomial[p +, ] p + Sum Biomial[p +, j] BeroulliB[j] p+-j, {j,, p} ; Now sum powers of the first 0 atural umbers, from power to power 0: I[6]:= Table[sumpower[0, k], {k,, 0}] Out[6]= {55, 385, 305, 5 333, 0 85, , , , , } For example = 385 ad = 305. Now use the summatio 0 k p for p 0 I[7]:= Table[Sum[j p, {j, 0}], {p,, 0}] Out[7]= {55, 385, 305, 5 333, 0 85, , , , , } Thus we have agreemet. A. B. C. D. Refereces Gamma, Julia Havil, Priceto Uiversity Press, 003 CRC Cocise Ecyclopedia of Mathematics, Eric W. Weisstei, Chapma ad Hall/CRC, 003 Special Fuctios, George E, Adrews, Richard Askey, Raja Roy, Cambridge Uiversity Press, 999 The sum of squares, Gagarie Yaikhom accessed December 04.