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.