360
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
The left hand side tends by definition to 10g(1 +x) as p increases indefinitely;
now the factors 1 - l/p, 1 - 1/2p, etc. figuring in (2) tend to 1. It follows
"obviously", as Halley, the comet man, and Euler remarked, that
But here we have a passage to the limit in a sum of infinitely many terms.
To justify this, we put
(12.3) up(n) = (_1)n(1 - l/p)(1 - 1/2p) ... (1 - l/np)xn+l /(n + 1),
so that
(12.4)
n
We have now to deal with a series whose terms are functions of an integer p, in other words, what we called a "sequence of series" in Chap. III, end
of nO 13. We know that, for all n,
(12.5)
lim up(n) = x n +! /(n + 1) = u(n)
p->oo
exists, that lup(n)1 ::::: lu(n)1 for all p and n, and we would like to be able to
deduce that
(12.6)
lim "up(n) = "u(n) = "
lim up(n).
v-co ~
~
~ p-+oo
n
n
n
The crucial point in the proof will be the fact that, for Ixl < 1, the series
L u p( n) is dominated by an absolutely convergent series independent of p,
namely v(n) = lu(n)l.
Theorem 9 (dominated convergence for series). Let Ln up(n),
pEN, be a sequence of series. Suppose that (i) up(n) tends to a limit u(n)
when p - t +00, (ii) (normal convergence) there exists a convergent series
with positive terms v( n) such that
(12.7)
for all p and n.
Then the series Lu(n) converges absolutely, and (6) holds.
This statement is in fact a particular case of Theorem 17 of Chap. III,
nO 13, on passing to the limit in a sequence of normally convergent series in
a variable t ETc C: here T = W, u(n, t) = ut(n) and t tends to a = +00.
But one can also proceed directly.
First, the absolute convergence of the series L u( n) is obvious since in the
limit we have lu(n)1 ::::: v(n) for all n. Now let s = L u(n) and sp = L up(n).
For all p and all integers N > 0 we have