124
II - Convergence: Discrete variables
The best way of seeing this is to associate to any x E R the positive
numbers x+ and x- defined as follows:
i+
sup(x,O) = x (if x ~ 0) or 0 (if x ~ 0),
x
sup( -x, 0) = 0 (if x ~ 0) or - x (if x ~ 0),
whence
On applying this stratagem to the terms of a real series L u( n) one exhibits
it as the difference of two series with positive terms whose sum is the series
L lu(n)l; since u(n)+ and u(n)- are ~ lu(n)l, it is clear that the lu(n)1 series
converges if and only if these two series with positive terms do so too. For a
series of complex terms, one can apply the method to the real and imaginary
parts v(n) and w(n) of its terms; one then has
(14.1)
u(n) = v(n)+ - v(n)- + iw(n)+ - iw(n)-,
four series with positive terms, which all converge if and only if the series of
the u( n) is absolutely convergent. It is clear that, more generally, any linear
combination of absolutely convergent series is absolutely convergent, since
la + b + c + ... 1 ~ lal + Ibl + Icl + ....
An absolutely convergent series is thus convergent, and since we have
commutativity for each of series v(n)+, etc. appearing in (1), we have it for
the given series. Furthermore
(14.2)
II: u(n)1 ~ I: lu(n)1
as one sees in passing to the limit in the analogous inequality for the partial
sums of the two sides.
Example 1. The exponential series
00
(14.3)
exp(z) = I: zn In! = I: z[nl
n=O
converges absolutely for all z E C and its sum is a continuous function of z.
We showed in nO 10, example 1, that it converges for z > 0, whence
absolute convergence in the general case since
The continuity of the function exp is an immediate result of the general
theorems of nO 19 on power series, but we shall need this earlier, so we shall
II - Convergence: Discrete variables
The best way of seeing this is to associate to any x E R the positive
numbers x+ and x- defined as follows:
i+
sup(x,O) = x (if x ~ 0) or 0 (if x ~ 0),
x
sup( -x, 0) = 0 (if x ~ 0) or - x (if x ~ 0),
whence
On applying this stratagem to the terms of a real series L u( n) one exhibits
it as the difference of two series with positive terms whose sum is the series
L lu(n)l; since u(n)+ and u(n)- are ~ lu(n)l, it is clear that the lu(n)1 series
converges if and only if these two series with positive terms do so too. For a
series of complex terms, one can apply the method to the real and imaginary
parts v(n) and w(n) of its terms; one then has
(14.1)
u(n) = v(n)+ - v(n)- + iw(n)+ - iw(n)-,
four series with positive terms, which all converge if and only if the series of
the u( n) is absolutely convergent. It is clear that, more generally, any linear
combination of absolutely convergent series is absolutely convergent, since
la + b + c + ... 1 ~ lal + Ibl + Icl + ....
An absolutely convergent series is thus convergent, and since we have
commutativity for each of series v(n)+, etc. appearing in (1), we have it for
the given series. Furthermore
(14.2)
II: u(n)1 ~ I: lu(n)1
as one sees in passing to the limit in the analogous inequality for the partial
sums of the two sides.
Example 1. The exponential series
00
(14.3)
exp(z) = I: zn In! = I: z[nl
n=O
converges absolutely for all z E C and its sum is a continuous function of z.
We showed in nO 10, example 1, that it converges for z > 0, whence
absolute convergence in the general case since
The continuity of the function exp is an immediate result of the general
theorems of nO 19 on power series, but we shall need this earlier, so we shall