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414 Notes on Explorations

sums. The paper C. Kelly, An algorithm for sums of integer powers, Math.

Mug. 57 (1984)) 296-297 gives an elementary derivation of

l+e( m~1)(lt+2X+...+n’)=(n+l)m+1.
k0
A simple recursive technique baaed on the lemma:
n
if c a,=~,, then c ra, = (n + l)s, - 2 s,
r=l f-=1 r=l
can be found in D. Sullivan, The sums of powers of integers (note 71.23))
Math. Gut. 71 (1987)) 144-146.
An application of matrices to the evaluation of power sums appears in
A.W.F. Edwards, Sums of powers of integers: a little of the history, Math.
Ga.zette 66 (1982), 22-28; and in A.W.F. Edwards, A quick route to sums
of powers, Amer. Math. Monthly 93 (1986), 451-455. A discussion of ap-
proximations for the sums is found in B.L. Burrows & R.F. Talbot, Sums
of powers of integers, Amer. Math. Monthly 91 (1984), 394-403.
Let &(n) be the sum of the first n squares, and for k 2 1, let Sk+l(n) =
Sk(l) + &(2) +... + Sk(n). A s tr aightforward induction argument shows
that
(k + 2)!&(n) = n(n + l)(n + 2)... (n + k)(2n + k).

This result is generalized in Problem 4380 in Amer. Math. Monthly 57
(1950)) 119; 58 (1951) 429.

E.58. Even though f,,(t) interpolates more and more values of Itl, in fact
link,, f,,(t) = ItI is true only for t = -1, 0, 1. For a reference to this fact,
see page 37 of G.G. Lorentz, Approximation of Functions, (Holt, Rinehart
& Winston, 1966).
This phenomenon occurs for other functions as well. For example, if
pn(t) interpolates (1 + t2)-l on [-5,5] at n + 1 equally spaced points, then
lim,, pn(t) = (1 + t2)-l when ItI does not exceed approximately 3.63.
Otherwise, the sequence diverges. A paper which analyses this rather subtle
issue is James F. Epperson, On the Runge example, Amer. Math. Monthly
94 (1987)) 329-341.

E.61. In general

is the sum of positive polynomials. For this and related results, consult
Sections 2.18-2.23 of G.H. Hardy, J.E. Littlewood & G. Polya, Inequalities,
(Cambridge, 1964).
David Hilbert (1862-1942) considered the following problem: Let f(x)
be a polynomial of degree n with m variables for which f(x) g 0 for all
real vectors x. Is it true that f(x) is the sum of squares of fimtely many
real polynomials? He showed that the answer is affirmative in the cases: