§ 1. Convergent sequences and series
85
obtained by dividing the relation
1 + 1/2 + 1/2 2 + 1/2 3 + ... = 2,
itself obtained on putting q = 1/2 in the sum (6.4) of the geometric series,
by 1, 3, 5, ... Having done this, Bernoulli adds these relations side-by-side as
one might do with a finite number of finite sums. One obtains the harmonic
series L 1/ n for the left hand side for the reason that any integer n can be
written in one and only one way as the product of a power of 2 and of an odd
number, and so appears once and only once among the left hand side terms.
By this ingenious procedure one finds the formula
1 + 1/2 + 1/3 + ... = 2(1 + 1/3 + 1/5 + ... )
whence "evidently"
1/2 + 1/4 + 1/6 + ... = 1 + 1/3 + 1/5 + ...
despite the fact that each term of the left hand side is strictly less than the
corresponding term of the right hand side. One might push the paradox even
further, substituting the right hand side into the left hand side; one would
thus obtain the formula
1/1.2 + 1/3.4 + 1/5.6 + ... = 0,
particularly miraculous since the terms of the left hand side are all > o. We
will see in nO 18, Corollary of Theorem 13, how one can justify this type of
operation, subject to hypotheses not satisfied in the preceding case.
One would be wrong to laugh at the Bernoullis. Even if, like the majority
of their contemporaries, they evinced an excessive penchant for virtuosity,
they did not have behind them three centuries of mathematicians who had
totally eliminated the difficulties inherent in the conception and use of series;
they were in process of inventing the subject starting from nothing or nearly
so. This protestant family, which left Anvers for Frankfort and then Bale
when the henchmen of the supercatholic Philippe II put down the revolt of
the Low Countries at the end of the XVlth century, produced eight mathematicians - the two brothers Jakob (1654-1705) and Johann (1667-1748),
the son Nikolaus (1687-1759) of a brother of these two, three sons of Johann,
Nikolaus II (1695-1726), Daniel (1700-1782) and Johann II (1710-1790) and
two sons of his, Johann III (1744-1807) and Jakob II (1759-1789) - well
known or famous, without speaking of their activities as physicists, jurists,
doctors, hellenists, etc.; see their notices in the DSB. The whole XVIllth
century calculated like them, notably Euler, the Bach of mathematics, student of Johann, and we shall yet see Fourier obtain prodigious results around
1807, by using series much more outrageously divergent than those of the
Bernoullis.
85
obtained by dividing the relation
1 + 1/2 + 1/2 2 + 1/2 3 + ... = 2,
itself obtained on putting q = 1/2 in the sum (6.4) of the geometric series,
by 1, 3, 5, ... Having done this, Bernoulli adds these relations side-by-side as
one might do with a finite number of finite sums. One obtains the harmonic
series L 1/ n for the left hand side for the reason that any integer n can be
written in one and only one way as the product of a power of 2 and of an odd
number, and so appears once and only once among the left hand side terms.
By this ingenious procedure one finds the formula
1 + 1/2 + 1/3 + ... = 2(1 + 1/3 + 1/5 + ... )
whence "evidently"
1/2 + 1/4 + 1/6 + ... = 1 + 1/3 + 1/5 + ...
despite the fact that each term of the left hand side is strictly less than the
corresponding term of the right hand side. One might push the paradox even
further, substituting the right hand side into the left hand side; one would
thus obtain the formula
1/1.2 + 1/3.4 + 1/5.6 + ... = 0,
particularly miraculous since the terms of the left hand side are all > o. We
will see in nO 18, Corollary of Theorem 13, how one can justify this type of
operation, subject to hypotheses not satisfied in the preceding case.
One would be wrong to laugh at the Bernoullis. Even if, like the majority
of their contemporaries, they evinced an excessive penchant for virtuosity,
they did not have behind them three centuries of mathematicians who had
totally eliminated the difficulties inherent in the conception and use of series;
they were in process of inventing the subject starting from nothing or nearly
so. This protestant family, which left Anvers for Frankfort and then Bale
when the henchmen of the supercatholic Philippe II put down the revolt of
the Low Countries at the end of the XVlth century, produced eight mathematicians - the two brothers Jakob (1654-1705) and Johann (1667-1748),
the son Nikolaus (1687-1759) of a brother of these two, three sons of Johann,
Nikolaus II (1695-1726), Daniel (1700-1782) and Johann II (1710-1790) and
two sons of his, Johann III (1744-1807) and Jakob II (1759-1789) - well
known or famous, without speaking of their activities as physicists, jurists,
doctors, hellenists, etc.; see their notices in the DSB. The whole XVIllth
century calculated like them, notably Euler, the Bach of mathematics, student of Johann, and we shall yet see Fourier obtain prodigious results around
1807, by using series much more outrageously divergent than those of the
Bernoullis.