126
II - Convergence: Discrete variables
measured by the mysterious number 21l", and other things too, particularly
from the theory of the integration. Clearly the founders of analysis did not
bother themselves with such rigour, quite unachievable in their time, and
this allowed them to progress. But we are no longer in the xvn th century,
and the best reply to these questions will be, as we shall do in nO 14 of
Chap. IV, to define the functions sinx and cosx by their power series and
then to deduce the elementary properties which we all expect: the addition
formulae, derivatives, relation cos 2 x + sin 2 x = 1, number 1l", etc.
Example 3. Consider the series Lzn/n2. Since lunl ::; 1/n 2 if Izl ::; 1 and
since the series L l/n 2 converges, the series L lunl satisfies the hypothesis of
Theorem 4 at least as well as the second does. It is thus absolutely convergent
for Izl ::; 1. We shall show later, with the help of certain simple criteria, that
it diverges for I z I > 1.
Example 4. Consider a power series L anz n and suppose that it converges
absolutely for z = u. Since lanznl ::; lanunl for Izl ::; lui, one can conclude
that the series again converges absolutely for Izl ::; lui. The fundamental
concept of radius of convergence of a power series, due to Cauchy, is obtained
by a similar argument. Consider the set E of numbers r ~ 0 for which the
sequence with general term lanlr n is bounded; denote by R either the least
upper bound of E if E is bounded, or the symbol +00 if it is not (nO 17).
Then the series L anz n converges absolutely for Izl < R and diverges for
Izl > R (one cannot say anything a priori as to what happens at the points
of the circumference Izl = R: anything is possible).
The second point is obvious since then the terms of the series are not
bounded, so do not tend to O. To establish the first, one notes that, by
definition of a least upper bound, there exists an r E E such that Izl < r < R,
whence Izl = qr with q < 1; putting M = sup (Ianlr n ), one then has lanznl =
qn lanr n I ::; M qn, whence absolute convergence, since q < 1.
These small calculations show a little more: the coefficients of a power
series with radius of convergence R > 0 cannot increase faster than a geometric progression. In other words, there are always positive constants M
and q such that
(14.5)
or, in the language of nO 3,
n - t +00.
This is obvious since, for all r < R, the sequence (anrn) is bounded (and even
tends to zero), so that it suffices to take q = l/r to obtain (5). Conversely, if
the an satisfy the relation (5), one has R > 0 since then the series converges
absolutely for qlzl < 1. The existence of a relation (5) thus characterises
power series with radius of convergence> 0 or, as one calls them dangerously,
the convergent power series.
II - Convergence: Discrete variables
measured by the mysterious number 21l", and other things too, particularly
from the theory of the integration. Clearly the founders of analysis did not
bother themselves with such rigour, quite unachievable in their time, and
this allowed them to progress. But we are no longer in the xvn th century,
and the best reply to these questions will be, as we shall do in nO 14 of
Chap. IV, to define the functions sinx and cosx by their power series and
then to deduce the elementary properties which we all expect: the addition
formulae, derivatives, relation cos 2 x + sin 2 x = 1, number 1l", etc.
Example 3. Consider the series Lzn/n2. Since lunl ::; 1/n 2 if Izl ::; 1 and
since the series L l/n 2 converges, the series L lunl satisfies the hypothesis of
Theorem 4 at least as well as the second does. It is thus absolutely convergent
for Izl ::; 1. We shall show later, with the help of certain simple criteria, that
it diverges for I z I > 1.
Example 4. Consider a power series L anz n and suppose that it converges
absolutely for z = u. Since lanznl ::; lanunl for Izl ::; lui, one can conclude
that the series again converges absolutely for Izl ::; lui. The fundamental
concept of radius of convergence of a power series, due to Cauchy, is obtained
by a similar argument. Consider the set E of numbers r ~ 0 for which the
sequence with general term lanlr n is bounded; denote by R either the least
upper bound of E if E is bounded, or the symbol +00 if it is not (nO 17).
Then the series L anz n converges absolutely for Izl < R and diverges for
Izl > R (one cannot say anything a priori as to what happens at the points
of the circumference Izl = R: anything is possible).
The second point is obvious since then the terms of the series are not
bounded, so do not tend to O. To establish the first, one notes that, by
definition of a least upper bound, there exists an r E E such that Izl < r < R,
whence Izl = qr with q < 1; putting M = sup (Ianlr n ), one then has lanznl =
qn lanr n I ::; M qn, whence absolute convergence, since q < 1.
These small calculations show a little more: the coefficients of a power
series with radius of convergence R > 0 cannot increase faster than a geometric progression. In other words, there are always positive constants M
and q such that
(14.5)
or, in the language of nO 3,
n - t +00.
This is obvious since, for all r < R, the sequence (anrn) is bounded (and even
tends to zero), so that it suffices to take q = l/r to obtain (5). Conversely, if
the an satisfy the relation (5), one has R > 0 since then the series converges
absolutely for qlzl < 1. The existence of a relation (5) thus characterises
power series with radius of convergence> 0 or, as one calls them dangerously,
the convergent power series.