84
II - Convergence: Discrete variables
whence the relation s = 1 + s for the "sum" s of the series. Immediately
Jakob Bernoulli, his elder brother, observed that the partial sum
l/(a + 1) + ... + 1/a 2
has a 2 - a terms all greater than 1/a 2 , so has a value greater than 1 - l/a,
whence it follows that l/a + ... + 1/a 2 > 1. From here it is easy to extract
from the harmonic series groupings of terms greater than an arbitrary integer. Jakob Bernoulli observed at that point that a series whose terms tend
to 0 - a condition clearly necessary for convergence since Un = Sn - Sn-l is
the difference of two sequences which tend to the same limit - can still be
divergent.
The fact that he had revealed the absurd marvels of the harmonic series
did not prevent Jakob, several years later, from baldly putting
A = 1/1 + 1/2 + 1/3 + ... ,
and then deducing that
A-I - 1/2 = 1/3 + 1/4 + ...
and then, by subtraction, that
3/2 = 2/1.3 + 2/2.4 + 2/3.5 + ... ,
which provided him a "new proof", this time correct, of a formula obtained
by Leibniz in 1682:
1/1.3 + 1/2.4 + 1/3.5 + ... = 3/4.
Similarly, putting E = 1/1 + 1/3 + 1/5 + ... (the series diverges), whence of
course E - 1 = 1/3 + 1/5 + ... , he obtains by difference and division by 2
another correct formula of Leibniz':
1/1.3 + 1/3.5 + 1/5.7 + ... = 1/2.
These calculations are meaningless. The usual rules of algebra were developed for calculating finite sums, i.e. consisting of only a finite number of
terms; it is sometimes legitimate to apply them to convergent series, and almost always, as we shall see later, to the absolutely convergent series which
we shall introduce in n° 15, not because they are obvious, but because the
mathematicians of the XIXth century have proved the indispensable general
theorems.
The same Jakob Bernoulli would give another precarious example in 1692.
He starts from the relations
1/1 + 1/2 + 1/4 + 1/8 + .. .
2/1,
1/3 + 1/6 + 1/12 + 1/24 + . . .
2/3
1/5 + 1/10 + 1/20 + 1/40 + . ..
2/5
II - Convergence: Discrete variables
whence the relation s = 1 + s for the "sum" s of the series. Immediately
Jakob Bernoulli, his elder brother, observed that the partial sum
l/(a + 1) + ... + 1/a 2
has a 2 - a terms all greater than 1/a 2 , so has a value greater than 1 - l/a,
whence it follows that l/a + ... + 1/a 2 > 1. From here it is easy to extract
from the harmonic series groupings of terms greater than an arbitrary integer. Jakob Bernoulli observed at that point that a series whose terms tend
to 0 - a condition clearly necessary for convergence since Un = Sn - Sn-l is
the difference of two sequences which tend to the same limit - can still be
divergent.
The fact that he had revealed the absurd marvels of the harmonic series
did not prevent Jakob, several years later, from baldly putting
A = 1/1 + 1/2 + 1/3 + ... ,
and then deducing that
A-I - 1/2 = 1/3 + 1/4 + ...
and then, by subtraction, that
3/2 = 2/1.3 + 2/2.4 + 2/3.5 + ... ,
which provided him a "new proof", this time correct, of a formula obtained
by Leibniz in 1682:
1/1.3 + 1/2.4 + 1/3.5 + ... = 3/4.
Similarly, putting E = 1/1 + 1/3 + 1/5 + ... (the series diverges), whence of
course E - 1 = 1/3 + 1/5 + ... , he obtains by difference and division by 2
another correct formula of Leibniz':
1/1.3 + 1/3.5 + 1/5.7 + ... = 1/2.
These calculations are meaningless. The usual rules of algebra were developed for calculating finite sums, i.e. consisting of only a finite number of
terms; it is sometimes legitimate to apply them to convergent series, and almost always, as we shall see later, to the absolutely convergent series which
we shall introduce in n° 15, not because they are obvious, but because the
mathematicians of the XIXth century have proved the indispensable general
theorems.
The same Jakob Bernoulli would give another precarious example in 1692.
He starts from the relations
1/1 + 1/2 + 1/4 + 1/8 + .. .
2/1,
1/3 + 1/6 + 1/12 + 1/24 + . . .
2/3
1/5 + 1/10 + 1/20 + 1/40 + . ..
2/5