§2. Absolutely convergent series
135
These traditional criteria are so useful that one is often tempted to believe
that a "small" modification of the hypotheses of Theorem 10 will be of little
importance, as if a theorem obeyed the same laws of stability as the pendulum
of a clock. In reality, a well constructed theorem resembles a pendulum in
upwards vertical equilibrium: a small impulse and it runs away. Examples:
(a) the series l/n: the d'Alembert ratio, n/(n + 1), tends to 1 and the
series diverges;
(b) the series 1/n 2 : the ratio, n 2 /(n + 1)2, tends to 1 and the series
converges;
( c) the series
obtained from the preceding by interchanging the terms in pairs: the d' Alembert
ratio again tends to 1 but is alternately > 1 and < 1, and the series converges
(compare its partial sums to those of the series 1/n 2 , or apply nO 11 bluntly).
(d) the series sin n /2 n : this is absolutely convergent since dominated by
the series 1/2n, but the d'Alembert ratio, namely Isin(n + 1)/2sinnl, does
not tend to any limit and, in fact, oscillates randomly between 0 and +00.
There are, of course, lots of more subtle criteria, applicable to the case
where the d'Alembert ratio tends to 1 through values < 1 (the series is
evidently divergent if it is > 1 for n large). The most famous, due to Gauss,
assumes a relation of the form
when n ~ +00;
if the Un are positive, the series converges for s > 1 and diverges for s ~ 1.
We shall return to this in Chap. VI.
For the moment we content ourselves with illustrating the theorem for
Newton's binomial series
(16.8)
1 + sz + s(s - 1)z2/2! +
+ s(s - l)(s - 2)z3/3! + ... = ~ (~)zn,
where z and s are complex and where the notation used for the coefficient of
zn is self-explanatory. It reduces to the polynomial expansion of (1 + z)S for
sEN and it was in extrapolating this case and that where 2s is integer that
Newton was led to his series, by a process more akin to divination than to
standard mathematics. Here U3/U2 = (s - 2)z/3, and, more generally,
Un+dun = (s - n)z/(n + 1).
The ratio thus tends to -z, whence it follows that Newton's series converges
absolutely for Izl < 1 and diverges for Izl > 1. And for Izl = I? More difficult;
see Chap. VI, hypergeometric series.
135
These traditional criteria are so useful that one is often tempted to believe
that a "small" modification of the hypotheses of Theorem 10 will be of little
importance, as if a theorem obeyed the same laws of stability as the pendulum
of a clock. In reality, a well constructed theorem resembles a pendulum in
upwards vertical equilibrium: a small impulse and it runs away. Examples:
(a) the series l/n: the d'Alembert ratio, n/(n + 1), tends to 1 and the
series diverges;
(b) the series 1/n 2 : the ratio, n 2 /(n + 1)2, tends to 1 and the series
converges;
( c) the series
obtained from the preceding by interchanging the terms in pairs: the d' Alembert
ratio again tends to 1 but is alternately > 1 and < 1, and the series converges
(compare its partial sums to those of the series 1/n 2 , or apply nO 11 bluntly).
(d) the series sin n /2 n : this is absolutely convergent since dominated by
the series 1/2n, but the d'Alembert ratio, namely Isin(n + 1)/2sinnl, does
not tend to any limit and, in fact, oscillates randomly between 0 and +00.
There are, of course, lots of more subtle criteria, applicable to the case
where the d'Alembert ratio tends to 1 through values < 1 (the series is
evidently divergent if it is > 1 for n large). The most famous, due to Gauss,
assumes a relation of the form
when n ~ +00;
if the Un are positive, the series converges for s > 1 and diverges for s ~ 1.
We shall return to this in Chap. VI.
For the moment we content ourselves with illustrating the theorem for
Newton's binomial series
(16.8)
1 + sz + s(s - 1)z2/2! +
+ s(s - l)(s - 2)z3/3! + ... = ~ (~)zn,
where z and s are complex and where the notation used for the coefficient of
zn is self-explanatory. It reduces to the polynomial expansion of (1 + z)S for
sEN and it was in extrapolating this case and that where 2s is integer that
Newton was led to his series, by a process more akin to divination than to
standard mathematics. Here U3/U2 = (s - 2)z/3, and, more generally,
Un+dun = (s - n)z/(n + 1).
The ratio thus tends to -z, whence it follows that Newton's series converges
absolutely for Izl < 1 and diverges for Izl > 1. And for Izl = I? More difficult;
see Chap. VI, hypergeometric series.