236
III - Convergence: Continuous variables
We remark that if the partial sums of the series Ul + U2 + ... are bounded
then so are, for all p, those of the series up + Up+l + ... since they differ from
the preceding by the number Ul + ... + Up-I, fixed for p given; the preceding
calculations thus apply to the "truncated" series upvp + ... even if the Vn ~ 0
decrease without necessarily tending to 0, a hypothesis which serves only to
pass from (2) to Cauchy's criterion for L UnVn . Putting
(11.3)
Mp = sup Iup + ... + unl
n~p
and applying the particular case p = 1 of (2) to the truncated series we thus
find
(11.4)
For given p this estimate is valid for any q > p, so one can pass to the limit
and, subject to Dirichlet's hypotheses, obtain the inequality
(11.5)
I L unvnl ~ 2Mpvp.
n~p
This can be used to prove the uniform convergence of certain series of functions:
Corollary 1 (AbeI 22 ). Let L anx n be a power series with finite radius of
convergence R > 0, and converging for x = R. Then the series converges
uniformly on the interval [0, R] (and, in fact, on all compacta contained in
] - R, R]); its sum is continuous on ] - R, R].
Let us put x = Ry, Un = anRn and Vn = yn, with 0 ~ y ~ 1. The series
L Un being convergent, its partial sums are bounded; the Vn are positive and
decrease, even for y = 1. Since UnVn = anx n we have, by (5),
I L anxnl ~ 2MpYp where Mp = sup lapRP + ... + anRnl·
n~p
n~p
Since the series Un = anRn converges we have Iup + ... + unl < £ for p and
n large, and so Mp ~ £ for p > N; since yP ~ 1, we see that
p > N ==> I L anx n I ~ 2£ for all x E [0, RJ,
n~p
which means that the difference between the total sum and the partial sums
of the power series converges to 0 uniformly on [0, R], qed.
This result applies for example to the series x - x 2 /2 + x 3 /3 - . .. for
10g(1 + x), but the method used for this case in nO 8, example 4, requires
definitely less ingenuity than Dirichlet's Theorem or its Corollary.
22 There is an analogous result in the complex domain; see Remmert, Punktionentheorie 1, p. 94.
III - Convergence: Continuous variables
We remark that if the partial sums of the series Ul + U2 + ... are bounded
then so are, for all p, those of the series up + Up+l + ... since they differ from
the preceding by the number Ul + ... + Up-I, fixed for p given; the preceding
calculations thus apply to the "truncated" series upvp + ... even if the Vn ~ 0
decrease without necessarily tending to 0, a hypothesis which serves only to
pass from (2) to Cauchy's criterion for L UnVn . Putting
(11.3)
Mp = sup Iup + ... + unl
n~p
and applying the particular case p = 1 of (2) to the truncated series we thus
find
(11.4)
For given p this estimate is valid for any q > p, so one can pass to the limit
and, subject to Dirichlet's hypotheses, obtain the inequality
(11.5)
I L unvnl ~ 2Mpvp.
n~p
This can be used to prove the uniform convergence of certain series of functions:
Corollary 1 (AbeI 22 ). Let L anx n be a power series with finite radius of
convergence R > 0, and converging for x = R. Then the series converges
uniformly on the interval [0, R] (and, in fact, on all compacta contained in
] - R, R]); its sum is continuous on ] - R, R].
Let us put x = Ry, Un = anRn and Vn = yn, with 0 ~ y ~ 1. The series
L Un being convergent, its partial sums are bounded; the Vn are positive and
decrease, even for y = 1. Since UnVn = anx n we have, by (5),
I L anxnl ~ 2MpYp where Mp = sup lapRP + ... + anRnl·
n~p
n~p
Since the series Un = anRn converges we have Iup + ... + unl < £ for p and
n large, and so Mp ~ £ for p > N; since yP ~ 1, we see that
p > N ==> I L anx n I ~ 2£ for all x E [0, RJ,
n~p
which means that the difference between the total sum and the partial sums
of the power series converges to 0 uniformly on [0, R], qed.
This result applies for example to the series x - x 2 /2 + x 3 /3 - . .. for
10g(1 + x), but the method used for this case in nO 8, example 4, requires
definitely less ingenuity than Dirichlet's Theorem or its Corollary.
22 There is an analogous result in the complex domain; see Remmert, Punktionentheorie 1, p. 94.