82
II - Convergence: Discrete variables
strictly increasing sequence of integers PbP2, ... such that I8
v(n) = u(Pn) for all n.
For example, the sequence (1/n 2 ) is a subsequence of the sequence (lin).
If a sequence (un) converges to a limit u, then every subsequence of (un)
converges to u. With the notation above, one has Pn 2:: n for all n since
Pn > ... > PI 2:: 1; the relation
d(u,un ) < r for all n > N
then implies that d(u, vn ) < r for all n > N, qed.
This trivial result translates usefully into the language of series. To extract
a subsequence from the sequence of partial sums Sn of a series u(n) one
chooses as above a sequence of integers Pn and considers the sequence whose
terms are
s(pt} = u(l) + ... + U(PI),
etc. These are manifestly the partial sums of the series whose successive terms
are
VI = u(l) + ... + u(pt} ,
V3 = U(P2 + 1) + ... + U(P3), ...
in other words, of the series obtained by grouping the terms of the initial
series into blocks of PI, P2 - PI, P3 - P2, . . . terms as if dealing with a finite
sum. Theorem 1 shows that, if the initial series converges, so does the new,
the two series having the same sum. This is an extension of the associativity
of addition: one has
u(l) +u(2) +u(3) + ... = [u(l) + ... + U(PI)] + [U(PI + 1) + ... + U(P2)] + ...
as for finite sums so long as the left hand side converges. One should be
aware of the fact that though a series may become convergent after grouping
its terms, it does not follow that it was already so before this operation: the
series (1 - 1) + (1 -1) + ... has no merit in being convergent, and the series
1 - 1 + 1 - 1 + ... is divergent. This difficulty does not arise with series of
positive terms as we shall see in nO 12.
The preceding artifice can serve to prove the divergence of a series, for
example of the harmonic series
1 + 1/2 + 1/3 + ...
If it were indeed convergent, so would be the series
18 The functional expression u(n) shows that a subsequence is only a particular case
of the general concept of composition of maps: compose n ...... Pn and n ...... Un.
II - Convergence: Discrete variables
strictly increasing sequence of integers PbP2, ... such that I8
v(n) = u(Pn) for all n.
For example, the sequence (1/n 2 ) is a subsequence of the sequence (lin).
If a sequence (un) converges to a limit u, then every subsequence of (un)
converges to u. With the notation above, one has Pn 2:: n for all n since
Pn > ... > PI 2:: 1; the relation
d(u,un ) < r for all n > N
then implies that d(u, vn ) < r for all n > N, qed.
This trivial result translates usefully into the language of series. To extract
a subsequence from the sequence of partial sums Sn of a series u(n) one
chooses as above a sequence of integers Pn and considers the sequence whose
terms are
s(pt} = u(l) + ... + U(PI),
etc. These are manifestly the partial sums of the series whose successive terms
are
VI = u(l) + ... + u(pt} ,
V3 = U(P2 + 1) + ... + U(P3), ...
in other words, of the series obtained by grouping the terms of the initial
series into blocks of PI, P2 - PI, P3 - P2, . . . terms as if dealing with a finite
sum. Theorem 1 shows that, if the initial series converges, so does the new,
the two series having the same sum. This is an extension of the associativity
of addition: one has
u(l) +u(2) +u(3) + ... = [u(l) + ... + U(PI)] + [U(PI + 1) + ... + U(P2)] + ...
as for finite sums so long as the left hand side converges. One should be
aware of the fact that though a series may become convergent after grouping
its terms, it does not follow that it was already so before this operation: the
series (1 - 1) + (1 -1) + ... has no merit in being convergent, and the series
1 - 1 + 1 - 1 + ... is divergent. This difficulty does not arise with series of
positive terms as we shall see in nO 12.
The preceding artifice can serve to prove the divergence of a series, for
example of the harmonic series
1 + 1/2 + 1/3 + ...
If it were indeed convergent, so would be the series
18 The functional expression u(n) shows that a subsequence is only a particular case
of the general concept of composition of maps: compose n ...... Pn and n ...... Un.