Chapter 1

Preliminary Notions

1.1. Approximating a sum by an integral

In certain situations, it is useful to replace a sum by an integral. The

following result shows when and how this can be done.

Proposition 1.1. Let a, b ∈ N with a b and let f : [a, b] → R be a

monotonic function on [a, b]. There exists a real number θ = θ(a, b) such

that −1 ≤ θ ≤ 1 and such that

(1.1)

an≤b

f(n) =

b

a

f(t) dt + θ(f(b) − f(a)).

Proof. Indeed, assume that f is decreasing, in which case, using a geometric

approach, it is easy to see that

0

b

a

f(t) dt −

an≤b

f(n)

a≤n≤b−1

f(n) −

an≤b

f(n) = f(a) − f(b),

from which (1.1) follows easily. If on the other hand, f is increasing, the

same type of argument yields (1.1).

One can use this result to estimate log n!. Indeed, setting f(n) = log n

in (1.1), one obtains

log n! =

n

i=1

log i =

n

1

log t dt + θ(log n − log 1) = n log n − n + O(log n),

thus providing a fairly good approximation for log n!. A better estimate is

proved in Section 1.9.

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http://dx.doi.org/10.1090/gsm/134/01