§2. Absolutely convergent series
127
"Dangerously" since there are those who believe that only series which
converge for any z E C should be called convergent. This is not the case; one
demands only that they should not diverge for every z i 0, which is quite
different. These series appear everywhere in classical analysis. The others
never, because one can do nothing with them; all the same, see the "formal
series" of nO 22.
It is obvious that in the interior of its circle of convergence the sum
J(z) = L: anz n of a power series is the limit of a sequence of polynomials
in z, namely the partial sums of the series. One can make this result more
specific by working within a disc of radius r strictly smaller than the radius of
convergence R of the series. Now Izn+HPI ~ Izln+1 r p for any p ~ 0, whence
since the sum L: p lan+plr P = M is convergent - it differs from the series
L: laprPI only by some terms at the beginning and a factor rn -, one finds
the inequality
(14.6)
In the notation of n° 3, this implies
(14.7)
but (6) is a little more specific because it is valid not only for r "small",
but even for all r < R (and so for all finite r if R = +00, as in the case of
the series for exp for example). The constant M of (6) depends on n and on
r; it is generally impossible to find an M for which (6) will be valid in the
whole disc of convergence: this is already false for the series L: zn. See nO 8
of Chap. III.
15 - Unconditional convergence: general case
After this incursion into the classical - the concept of absolute convergence
does not however antedate Cauchy and above all it was Weierstrass who
exploited it systematically - we come to that of unconditional convergence
for a sum of complex numbers L: u( i) where the index i varies in an arbitrary
countable set I. It allows us to elucidate the concept of absolutely convergent
series and, indeed, reduces to this; but it is indispensable for other reasons.
First we show how one can define it in a way similar to that of nO 12.
Suppose first that the u(i) ~ 0. The sum 38 s = s(l) is, by definition, the
least upper bound of the unordered partial sums
38 We use the notation s(F) for any subset F of I, finite or not, under the clear
condition that the sum has a meaning or, in the case of a sum of positive terms,
agreeing that s(F) = +00 if the sum does not converge unconditionally.
127
"Dangerously" since there are those who believe that only series which
converge for any z E C should be called convergent. This is not the case; one
demands only that they should not diverge for every z i 0, which is quite
different. These series appear everywhere in classical analysis. The others
never, because one can do nothing with them; all the same, see the "formal
series" of nO 22.
It is obvious that in the interior of its circle of convergence the sum
J(z) = L: anz n of a power series is the limit of a sequence of polynomials
in z, namely the partial sums of the series. One can make this result more
specific by working within a disc of radius r strictly smaller than the radius of
convergence R of the series. Now Izn+HPI ~ Izln+1 r p for any p ~ 0, whence
since the sum L: p lan+plr P = M is convergent - it differs from the series
L: laprPI only by some terms at the beginning and a factor rn -, one finds
the inequality
(14.6)
In the notation of n° 3, this implies
(14.7)
but (6) is a little more specific because it is valid not only for r "small",
but even for all r < R (and so for all finite r if R = +00, as in the case of
the series for exp for example). The constant M of (6) depends on n and on
r; it is generally impossible to find an M for which (6) will be valid in the
whole disc of convergence: this is already false for the series L: zn. See nO 8
of Chap. III.
15 - Unconditional convergence: general case
After this incursion into the classical - the concept of absolute convergence
does not however antedate Cauchy and above all it was Weierstrass who
exploited it systematically - we come to that of unconditional convergence
for a sum of complex numbers L: u( i) where the index i varies in an arbitrary
countable set I. It allows us to elucidate the concept of absolutely convergent
series and, indeed, reduces to this; but it is indispensable for other reasons.
First we show how one can define it in a way similar to that of nO 12.
Suppose first that the u(i) ~ 0. The sum 38 s = s(l) is, by definition, the
least upper bound of the unordered partial sums
38 We use the notation s(F) for any subset F of I, finite or not, under the clear
condition that the sum has a meaning or, in the case of a sum of positive terms,
agreeing that s(F) = +00 if the sum does not converge unconditionally.