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II - Convergence: Discrete variables
We have seen above that a series L Un whose terms satisfy a relation of
the form Un = O(qn) with 0 < q < 1 is absolutely convergent. There are
two classical criteria, due respectively to d' Alembert, the man of the Encyclopaedia and of the Enlightenment, militant atheist, and to Cauchy, the
ultra-legitimist of the Polytechnique School, and ultra-catholic of the Restoration, who developed this quasimechanistic method of comparison. Since, in
practice, they frequently apply to the same series, readers of the left may
prefer d'Alembert, those of the right Cauchy:
Theorem 10. Let L Un be a series with complex terms. Suppose either that
the ratio IUn+1/unl (d'Alembert), or that lunl 1 / n (Cauchy) tends to a limit q
as n -+ 00. Then the series is absolutely convergent if q < 1 and divergent if
q> 1.
If indeed lim Iun+dunl = q < 1 and if you choose a number q' such
that q < q' < 1, then Iun+dunl < q' for n large, say for n > p. Then
IUp+rl < q'IUp+r-ll < q,2Iup+r_21 < ... < q,rlupl for all r > 1 and thus
Iunl < Mqln for n large, with a constant M = lupl/q'P (put n = p+r). Since
q' < 1, the geometric series L q,r converges, whence the absolute convergence
of the given series. The argument is even simpler in the Cauchy case: one has
lunl 1 / n < q' for n large, i.e. Iunl < qln, and concludes as before.
If q > 1, this time one chooses q' so that 1 < q' < q. One obtains the same
inequalities, but with > signs instead of < signs and then, for n large, the
expressions of d'Alembert and of Cauchy are> q'. But then the inequality
Iunl > Mqln proves not only that the series diverges, but that Iunl increases
indefinitely, qed.
As examples, consider the following power series:
(a) L zn In!; here IUn+1/unl = Izl/(n + 1) as one line of calculation shows;
the ratio thus tends to 0 for any z, whence the absolute convergence of
the exponential series; the radius of convergence (nO 14, example 4) is
+00.
(b) Lzn/n s ; the d'Alembert ratio, equal to Izl/(l + l/n)B, tends to Izl;
therefore absolute convergence for Izl < 1 and divergence for Izi > 1; the
radius of convergence is R = 1. Note that the Cauchy expression, equal
to lzi.nl/ n , also tends to Izl (nO 5, example 8). Note also that, for Izi = 1,
one can say nothing, so that the Riemann series L l/n s does not come
within the scope of the very weak Theorem 10.
(c) L n 2 zn; the Cauchy expression, namely Izl(n 1 /n)2, again tends to Izl;
same conclusions.
(d) Ln!Zn; the d'Alembert ratio, namely (n + l)lzl, increases indefinitely
for z i- o. The series is always divergent, except of course for z = o. The
radius of convergence is zero.
II - Convergence: Discrete variables
We have seen above that a series L Un whose terms satisfy a relation of
the form Un = O(qn) with 0 < q < 1 is absolutely convergent. There are
two classical criteria, due respectively to d' Alembert, the man of the Encyclopaedia and of the Enlightenment, militant atheist, and to Cauchy, the
ultra-legitimist of the Polytechnique School, and ultra-catholic of the Restoration, who developed this quasimechanistic method of comparison. Since, in
practice, they frequently apply to the same series, readers of the left may
prefer d'Alembert, those of the right Cauchy:
Theorem 10. Let L Un be a series with complex terms. Suppose either that
the ratio IUn+1/unl (d'Alembert), or that lunl 1 / n (Cauchy) tends to a limit q
as n -+ 00. Then the series is absolutely convergent if q < 1 and divergent if
q> 1.
If indeed lim Iun+dunl = q < 1 and if you choose a number q' such
that q < q' < 1, then Iun+dunl < q' for n large, say for n > p. Then
IUp+rl < q'IUp+r-ll < q,2Iup+r_21 < ... < q,rlupl for all r > 1 and thus
Iunl < Mqln for n large, with a constant M = lupl/q'P (put n = p+r). Since
q' < 1, the geometric series L q,r converges, whence the absolute convergence
of the given series. The argument is even simpler in the Cauchy case: one has
lunl 1 / n < q' for n large, i.e. Iunl < qln, and concludes as before.
If q > 1, this time one chooses q' so that 1 < q' < q. One obtains the same
inequalities, but with > signs instead of < signs and then, for n large, the
expressions of d'Alembert and of Cauchy are> q'. But then the inequality
Iunl > Mqln proves not only that the series diverges, but that Iunl increases
indefinitely, qed.
As examples, consider the following power series:
(a) L zn In!; here IUn+1/unl = Izl/(n + 1) as one line of calculation shows;
the ratio thus tends to 0 for any z, whence the absolute convergence of
the exponential series; the radius of convergence (nO 14, example 4) is
+00.
(b) Lzn/n s ; the d'Alembert ratio, equal to Izl/(l + l/n)B, tends to Izl;
therefore absolute convergence for Izl < 1 and divergence for Izi > 1; the
radius of convergence is R = 1. Note that the Cauchy expression, equal
to lzi.nl/ n , also tends to Izl (nO 5, example 8). Note also that, for Izi = 1,
one can say nothing, so that the Riemann series L l/n s does not come
within the scope of the very weak Theorem 10.
(c) L n 2 zn; the Cauchy expression, namely Izl(n 1 /n)2, again tends to Izl;
same conclusions.
(d) Ln!Zn; the d'Alembert ratio, namely (n + l)lzl, increases indefinitely
for z i- o. The series is always divergent, except of course for z = o. The
radius of convergence is zero.