§l. Convergent sequences and series
77
plane ]R2, and suppose that you are interested in the sum of inverses of the
kth powers of the distances from the origin to the points of the lattice, i.e.
you wish to assign a meaning to the sum of numbers 1/(m 2 + n 2 }k/2, where
m and n are rational integers, not both zero. We know that 7.,2 is countable,
but there is no privileged, natural or obvious bijection of N onto 7.,2. This
poses, in this case, the problem of defining the "unordered" sum of the terms
of the series. We shall study this later (nO 11), but for the moment we confine
ourselves, maybe wrongly, to the traditional situation of a sum whose terms
are given in the form of an ordered sequence.
To obtain a simple series, or series for short, without explicit mention to
the contrary, one starts with a sequence (un) of complex numbers. Since the
total sum of the Un can reasonably be defined only through an approximation
procedure involving only the sums of a finite number of terms - the only ones
that we know at this stage of the exposition -, it is natural to consider the
numbers
81
Ul
82
Ul + U2
83
= Ul + U2 + U3
etc. One calls them the ordered partial sums, or partial sums for short when
there is no fear of confusion, of the series with general term Un, and one says
that this is convergent when lim 8n = 8, the sum of the series considered,
exists. Then one writes
00
8= LUn'
n=1
or 8 = Ul + U2 + ... ,
or simply 8 = L Un. SO, by definition,
(6.1)
8 = lim(ul + ... + Un).
The notation Ul + U2 + ... , as used by all the Founding Fathers, is now in
total desuetude, but the reader will maybe find it, at the beginning, easier
than the other.
As the convergence of a series reduces to that of a sequence, conversely
the relation
Un = Ul + (U2 - uI) + ... + (un - un-I)
reduces the convergence of a sequence to that of a series.
Some people seem to believe that it is contrary to the principles of sane
pedagogy to introduce series at the beginning of teaching analysis. At the risk
of traumatising the reader, let us observe that the nonterminating decimal
expansion
77
plane ]R2, and suppose that you are interested in the sum of inverses of the
kth powers of the distances from the origin to the points of the lattice, i.e.
you wish to assign a meaning to the sum of numbers 1/(m 2 + n 2 }k/2, where
m and n are rational integers, not both zero. We know that 7.,2 is countable,
but there is no privileged, natural or obvious bijection of N onto 7.,2. This
poses, in this case, the problem of defining the "unordered" sum of the terms
of the series. We shall study this later (nO 11), but for the moment we confine
ourselves, maybe wrongly, to the traditional situation of a sum whose terms
are given in the form of an ordered sequence.
To obtain a simple series, or series for short, without explicit mention to
the contrary, one starts with a sequence (un) of complex numbers. Since the
total sum of the Un can reasonably be defined only through an approximation
procedure involving only the sums of a finite number of terms - the only ones
that we know at this stage of the exposition -, it is natural to consider the
numbers
81
Ul
82
Ul + U2
83
= Ul + U2 + U3
etc. One calls them the ordered partial sums, or partial sums for short when
there is no fear of confusion, of the series with general term Un, and one says
that this is convergent when lim 8n = 8, the sum of the series considered,
exists. Then one writes
00
8= LUn'
n=1
or 8 = Ul + U2 + ... ,
or simply 8 = L Un. SO, by definition,
(6.1)
8 = lim(ul + ... + Un).
The notation Ul + U2 + ... , as used by all the Founding Fathers, is now in
total desuetude, but the reader will maybe find it, at the beginning, easier
than the other.
As the convergence of a series reduces to that of a sequence, conversely
the relation
Un = Ul + (U2 - uI) + ... + (un - un-I)
reduces the convergence of a sequence to that of a series.
Some people seem to believe that it is contrary to the principles of sane
pedagogy to introduce series at the beginning of teaching analysis. At the risk
of traumatising the reader, let us observe that the nonterminating decimal
expansion