§l. Convergent sequences and series
83
1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ...
obtained by grouping of 1,2,4,8, 16, ... terms in the initial series (one omits
the first term which clearly plays no role in the question). Now the first group
of terms has a value :::- 1/2, the second, being the sum of two terms greater
than 1/4, is the same, the third again, since it is the sum of four terms greater
than 1/8, etc. One thus finds, for the new series, partial sums successively
greater than 1/2, 1, 3/2, etc.; whence the divergence of the new series and
so of the harmonic series. This kind of argument will be generalised in nO 12
("Cauchy's condensation criterion").
There are many variants of the preceding proof; they have sometimes given
rise, historically, to spectacular errors 19 of a nature to commend prudence to
readers who are starting on the subject. A little after 1650, the Italian Pietro
Mengoli observed that one always has
1
1
1
3
--+-+-- >-'
n-1
n
n+1
n'
so, by grouping the terms of the harmonic series, one obtains
1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + ... > 1 + 3/3 + 3/6 + ...
= 1 + 1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + (1/8 + ...
> 1 + 1 + 3/3 + 3/6 + ... = 1 + 1 + 1 + (1/2 + 1/3 + 1/4) + ...
etc., which shows that the proposed sum s of the series is greater than any
integer. Instead of advertising his ingenuousness, Mengoli should have confined himself to observing that the first of his inequalities already provides
the rather fishy relation s > s + 1.
Forty years later, Johann Bernoulli used an analogous idea. He started
from the relation
1/1.2 + 1/2.3 + 1/3.4 + '" = 1,
obvious if one writes 20 it as
(1 - 1/2) + (1/2 - 1/3) + ... = 1,
and then remarked that
1/2 + 1/3 + 1/4 + ... = 1/1.2 + 2/2.3 + 3/3.4 + ...
= (1/1.2 + 1/2.3 + 1/3.4 + ... ) + (1/2.3 + 1/3.4 + ... ) + (1/3.4 + ... )
= 1 + (1 - 1/2) + (1- 1/2 - 1/6) + (1 - 1/2 - 1/6 - 1/12) + ...
= 1 + 1/2 + 1/3 + 1/4 + ... ,
19 I find them in Cantor, Vorlesungen ... , Vol. III, particularly pp. 94-96.
20 The relation in question is obvious if one calculates as with a finite sum since
all the terms apart from the first "visibly" cancel in pairs. But the correct proof
consists of remarking that the sum of the first terms n, namely 1 - lin, tends
to 1 as lin tends to O.
83
1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + ...
obtained by grouping of 1,2,4,8, 16, ... terms in the initial series (one omits
the first term which clearly plays no role in the question). Now the first group
of terms has a value :::- 1/2, the second, being the sum of two terms greater
than 1/4, is the same, the third again, since it is the sum of four terms greater
than 1/8, etc. One thus finds, for the new series, partial sums successively
greater than 1/2, 1, 3/2, etc.; whence the divergence of the new series and
so of the harmonic series. This kind of argument will be generalised in nO 12
("Cauchy's condensation criterion").
There are many variants of the preceding proof; they have sometimes given
rise, historically, to spectacular errors 19 of a nature to commend prudence to
readers who are starting on the subject. A little after 1650, the Italian Pietro
Mengoli observed that one always has
1
1
1
3
--+-+-- >-'
n-1
n
n+1
n'
so, by grouping the terms of the harmonic series, one obtains
1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + ... > 1 + 3/3 + 3/6 + ...
= 1 + 1 + (1/2 + 1/3 + 1/4) + (1/5 + 1/6 + 1/7) + (1/8 + ...
> 1 + 1 + 3/3 + 3/6 + ... = 1 + 1 + 1 + (1/2 + 1/3 + 1/4) + ...
etc., which shows that the proposed sum s of the series is greater than any
integer. Instead of advertising his ingenuousness, Mengoli should have confined himself to observing that the first of his inequalities already provides
the rather fishy relation s > s + 1.
Forty years later, Johann Bernoulli used an analogous idea. He started
from the relation
1/1.2 + 1/2.3 + 1/3.4 + '" = 1,
obvious if one writes 20 it as
(1 - 1/2) + (1/2 - 1/3) + ... = 1,
and then remarked that
1/2 + 1/3 + 1/4 + ... = 1/1.2 + 2/2.3 + 3/3.4 + ...
= (1/1.2 + 1/2.3 + 1/3.4 + ... ) + (1/2.3 + 1/3.4 + ... ) + (1/3.4 + ... )
= 1 + (1 - 1/2) + (1- 1/2 - 1/6) + (1 - 1/2 - 1/6 - 1/12) + ...
= 1 + 1/2 + 1/3 + 1/4 + ... ,
19 I find them in Cantor, Vorlesungen ... , Vol. III, particularly pp. 94-96.
20 The relation in question is obvious if one calculates as with a finite sum since
all the terms apart from the first "visibly" cancel in pairs. But the correct proof
consists of remarking that the sum of the first terms n, namely 1 - lin, tends
to 1 as lin tends to O.