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7.10. OTHER CALCULUS PROBLEMS 195
7.10 Other Calculus Problems
7.10.1 Indefinite summation
The initial paper in this area was [Gos78]. Although not expressed that way in
this paper, the key idea [Kar81] is the following, akin to Definition 80.
Definition 87 A difference ring is a ring (Definition 6) equipped with an additional
unary operation, referred to as differentiation and written 10 with a prefix
δ, which satisfies three additional laws:
1. δ(f + g) = δf + δg;
2. δ(fg) = f(δg) + (δf)g + (δf)(δg);
3. δ(1) = 0.
A difference ring which is also a field (Defintion 13) is referred to as a difference
field.
One thinks of δ(f) as being f(n+1)−f(n). It is worth remarking the differences
with Definition 80: clause 3 is necessary, whereas previously it could be inferred
from the others, and clause 2 is different, essentially because (x 2 ) ′ = 2x but
δ(n 2 ) = (n + 1) 2 − n 2 = 2n + 1.
The process of formal (indefinite) summation ∑ is then that of inverting δ,
as ∫ is the prcess of inverting ′ . Just as ∫the rôle of dx in integration theory
is explained by the fact that x ′ = 1, i.e. 1 = x, so the traditional rôle of n
∑
in summation theory is explained by the fact that δn = 1, equivalently that
1 = n. Note that the two procedures of integration and summation are more
similar than normal notation would suggest: we are happy with ∫ x = 1 2 x2 , but
less so with ∑ n = 1 2n(n + 1), yet the two are essentially equivalent.
To further the analogy, we know that ∫ 1
x
cannot be expressed as a rational
∑
function of x, but is a new expression, known conventionally as log x. Similarly,
1
n
cannot be expressed as a rational function of n [Kar81, Example 16], but
is a new expression, known conventionally as H n , the n-th harmonic number.
Again, ∫ log x = x log x − x, and ∑ H n = nH n − n.
7.10.2 Definite Symbolic Summation
Definite summation might be thought to be related to indefinite summation in
the way that definite integration (Section 7.9) is to indefinite. There is certainly
the same problem of evaluating expressions from difference algebra at numerical
points. However, in practice we are more concerned with a subtly different class
of problems, where the limits of summation also figure in the summand.
Example 12 ([GKP94, Exercise 6.69], [Sch00b])
n∑
k 2 H n+k = 1 (
3 n n + 1 )
(n + 1)(2H 2n − H n ) − 1 2
36 n(10n2 + 9n − 1). (7.52)
k=1
10 In this text: notations differ.