Chapter 30

Sums of powers of natural

numbers

Notation

Sk(n):=1k+2k+···+nk.

Theorem 30.1(Bernoulli).Sk(n)is a polynomial innof degreek+1
without constant term. It can be obtained recursively as

Sk+1(n)=

∫

(k+1)Sk(n)dn+cn,

wherecis determined by the condition that the sum of the coefficients is

Examples

(1)S 3 (n)=1^3 +2^3 +···+n^3 =^14 n^2 (n+1)^2.
(2) Since 4 S 3 (n)=n^4 +2n^3 +n^2 ,wehave

S 4 (n)=

1

5

n^5 +

1

2

n^4 +

1

3

n^3 +cn,

wherec=1−

( 1

5 +

1

2 +

1

3

)

=− 301. Therefore,

14 +2^4 +···+n^4 =

1

5

n^5 +

1

2

n^4 +

1

3

n^3 −

1

30

n.