242
III - Convergence: Continuous variables
(13.1)
lim "u(n, t) = "lim u(n, t) = "u(n),
t--+a ~
~ t--+a
~
N
N
and further, the series E u( n) is absolutely convergent.
The existence of a convergent series with positive terms E v( n) such that
lu(n, t)1 :::; v(n) for all nand t shows that also, in the limit, lu(n)1 :::; v(n),
whence the absolute convergence of the series of the limits u(n).
As for the relation (1), one obtains it by observing that the sequence of
partial sums sn(t) = u(l, t) + ... + u(n, t) converges to s(t) uniformly on T
(n° 8, Theorem 9), so that (Theorem 16)
lim lim sn(t) = lim lim sn(t);
t-+a n--+oo
n-+oo t--+a
the left hand side, the limit for n - t 00 is simply, by definition, the total sum
s(t) of the series E u(n, t); the right hand side, the limit as t - t a is just the
partial sum Sn of the series of limits, so that the limit as n - t 00 is the total
sum s of this, qed.
We can also prove this theorem directly. First, the absolute convergence
of the series E u( n) is obvious as we have seen above; let us write s for its
sum. For all t and all pEN we then have
Is(t) - sl < L lu(n, t) - u(n)1 + L lu(n, t) - u(n)1 :::;
n>p
< L lu(n, t) - u(n)1 + 2 L v(n).
n$;p
n>p
Choose an r > o. For p sufficiently large, v(p + 1) + v(p + 2) + ... is < r, so
that the second term is < 2r for any t. Having chosen such a p, each of the
p terms of the first sum is < r /p for t sufficiently close to a, so that the said
sum is itself < r for t sufficiently close to a. Adding, the difference Is(t) - sl
is thus < 3r for all t E T sufficiently close to a, qed.
It is clear that if X is a subset of lR, and not bounded above, one can
also choose a = +00 in the preceding statements, respecting the usual conventions: replace "t sufficiently close to a" by "t sufficiently large" .
Since the set T of Theorem 17 is required only to be a subset of C - so
as. to give a meaning to the expression ''when t E T tends to a" -, one can
choose T = N. Instead of a series E u(n, t) whose general term depends on a
"parameter" t E T, one has a series u(n,p) whose general term depends on
the summation variable n and on an integer pEN. When one is passing to
a limit, and not summing over p, it is natural to consider that one is in the
presence of a "sequence of series" and in consequence to denote the general
term of series nO p by up(n). We shall state the result when we in need it
in Chap. IV, nO 12, but the reader should have no trouble in formulating it
now.
III - Convergence: Continuous variables
(13.1)
lim "u(n, t) = "lim u(n, t) = "u(n),
t--+a ~
~ t--+a
~
N
N
and further, the series E u( n) is absolutely convergent.
The existence of a convergent series with positive terms E v( n) such that
lu(n, t)1 :::; v(n) for all nand t shows that also, in the limit, lu(n)1 :::; v(n),
whence the absolute convergence of the series of the limits u(n).
As for the relation (1), one obtains it by observing that the sequence of
partial sums sn(t) = u(l, t) + ... + u(n, t) converges to s(t) uniformly on T
(n° 8, Theorem 9), so that (Theorem 16)
lim lim sn(t) = lim lim sn(t);
t-+a n--+oo
n-+oo t--+a
the left hand side, the limit for n - t 00 is simply, by definition, the total sum
s(t) of the series E u(n, t); the right hand side, the limit as t - t a is just the
partial sum Sn of the series of limits, so that the limit as n - t 00 is the total
sum s of this, qed.
We can also prove this theorem directly. First, the absolute convergence
of the series E u( n) is obvious as we have seen above; let us write s for its
sum. For all t and all pEN we then have
Is(t) - sl < L lu(n, t) - u(n)1 + L lu(n, t) - u(n)1 :::;
n>p
< L lu(n, t) - u(n)1 + 2 L v(n).
n$;p
n>p
Choose an r > o. For p sufficiently large, v(p + 1) + v(p + 2) + ... is < r, so
that the second term is < 2r for any t. Having chosen such a p, each of the
p terms of the first sum is < r /p for t sufficiently close to a, so that the said
sum is itself < r for t sufficiently close to a. Adding, the difference Is(t) - sl
is thus < 3r for all t E T sufficiently close to a, qed.
It is clear that if X is a subset of lR, and not bounded above, one can
also choose a = +00 in the preceding statements, respecting the usual conventions: replace "t sufficiently close to a" by "t sufficiently large" .
Since the set T of Theorem 17 is required only to be a subset of C - so
as. to give a meaning to the expression ''when t E T tends to a" -, one can
choose T = N. Instead of a series E u(n, t) whose general term depends on a
"parameter" t E T, one has a series u(n,p) whose general term depends on
the summation variable n and on an integer pEN. When one is passing to
a limit, and not summing over p, it is natural to consider that one is in the
presence of a "sequence of series" and in consequence to denote the general
term of series nO p by up(n). We shall state the result when we in need it
in Chap. IV, nO 12, but the reader should have no trouble in formulating it
now.