Normat 60:1, 1–6 (2012) 1
A note on series of positive numbers
Jorge Jiménez Urroz
Departamento de Matemática Aplicada IV
Universidad Politecnica de Catalunya
Barcelona, 08034, España
jjimenez@ma4.upc.edu
1 Introduction
The study of series appears everywhere in Analysis. And the first issue is to know
whether the series is convergent or not. Most of the times we need to appeal to
absolute convergence and, in this way, we end up by trying to understand series of
positive numbers. There are several criteria to decide if a series of positive numbers
is convergent or not, however most of them seems to have two similar characte-
ristics: first, they come, in one way or another, from the Comparison Principle. In
certain sense, one could think that this fact is limiting our study of convergence of
series of positive numbers. Second, none of them gives equivalent conditions. For
example D’Alambert’s, Cauchy’s or Raabe’s criteria fail when the corresponding
limit is 1.
Here we present another criterion which gives an equivalent condition for the
convergence of a series of positive numbers, which in fact does not come from the
Comparison Principle. This criterion came to me when I was trying to prove to the
students that the dual of L
2
is L
2
in an elementary way, now Corollay 7 below.
By gravitation law, I came needing to prove Theorem 2. Searching and asking to
experts, one could conclude that the result is not entirely known to the experts, even
though seems a nice an useful result as one can see for the applications included
here. Even though finally some partial references, included in the bibliography,
appeared in the process it was only the referee who mentioned to me the excellent
book by de la Vallée Poussin (7). In page 432 of this book there is a result presented
as Exercise 3, that contains basically all the results presented here, as well as the
results in (1) and (3), as particular cases. In the note I include both the Exercise of
de la Vallée Poussin’s book, and also the theorem that appeared naturally to me,
together with a short, selfcontained proof, and also several applications that enlight
the power of this criterion, which could be considered as part of the standard theory
of series of positive numbers.
2 Jorge Jiménez Urroz Normat 1/2012
Theorem 1 (De la Vallée Poussin) Let a
n
≥ 0 and
P
a
n
divergent. Suppose f(x)
is decreasing and lim
x→∞
f(x) = 0. Let S
N
=
P
k≤N
a
k
and F (x) =
R
x
0
f(t)dt.
Consider
S
f,1
=
X
f(S
n
)a
n
S
f,2
=
X
f(S
n−1
)a
n
.
Then,
1. If F (x) is bounded then S
f,1
is convergent.
2. If F (x) is unbounded then S
f,2
is divergent.
3. If a
n
is bounded, then S
f,1
and S
f,2
are both convergent or both divergent at
the same time.
Theorem 2 Let a
k
> 0 for k ≥ 0, and S
N
=
P
k≤N
a
k
. Then,
1.
P
a
k
converges if and only if
P
a
k
S
k
does.
2.
P
a
k
S
k
(log(S
k
+1))
2
is always convergent.
Remark. Note that in Theorem 1 the monotony condition on f(x) already gua-
rantees the existence of F (x). Also, the “difficult” implication of part (1) of the
Theorem 2 is already proved in a paper by Abel in 1828, (1). Dini in 1867 in (3)
improved his result and obtained the convergence of the series
P
n6=N
a
n
S
α
n
for α > 1.
However, their proofs, more involved than the showed here, lose enough so it is not
achieved the second part of Theorem 2. The result when
P
a
n
is convergent, is not
a consequence of Theorem 1. We give an example below.
2 Proofs.
Before proving the theorem, we should add some remarks. First let us say that
Theorem 2 is in fact a criterion for series of positive numbers. Indeed, otherwise it
could happen that
a
k
S
k
is not well defined for infinitely many k. But even if this is
not the case, one can not ensure the result. Let us for example consider a
1
= 1 and
a
k
= 3(−1/2)
k−1
for any k ≥ 2. Then
P
a
k
is convergent by Leibniz’s criterion.
However, for any k ≥ 2, S
k
=
P
k
j=1
a
j
=
1
3
a
k
, and so
P
a
k
S
k
is divergent. Also, we
should note that the second part of the theorem would not remain true by removing
1 from the logarithm. To see this, consider a
k
=
1
k(k+1)
. Then, S
k
= 1 −
1
k+1
,
| log S
k
| <
2
k+1
, and so
P
a
k
S
k
(log(S
k
))
2
>
1
4
P
1
1−
1
k+1
, is a divergent series. Notice
that in this case S
k
is convergent. Clearly this is the only case in which adding 1
to the argument of the logarithm is an important matter.
2.1 Proof of the Theorem 1.
To prove Parts (1) and (2), we note that, since f is decreasing,
f(S
n
)a
n
≤
Z
S
n
S
n−1
f(t)dt ≤ f(S
n−1
)a
n
Normat 1/2012 Jorge Jiménez Urroz 3
and so
X
n≤N
f(S
n
)a
n
≤
Z
S
N
0
f(t)dt ≤
X
n≤N
f(S
n−1
)a
n
.
The result follows by taking limits when N → ∞. To prove Part (3), observe that
0 ≤
X
n≤N
(f(S
n−1
) − f(S
n
)) a
n
≤ K
X
n≤N
(f(S
n−1
) − f(S
n
)) = K(f(0) − f(S
N
)),
whenever a
n
≤ K, by the monotonicity of f . Again, taking limits we find that
0 ≤ S
f,2
− S
f,1
≤ Kf(0) and the result follows.
2.2 Proof of the Theorem 2.
If S =
P
a
k
is convergent, both results in the theorem are trivial. Indeed, note
that in this case S
k
> S − ε for any ε > 0 and k sufficiently large depending on
ε. Then, we assume S
N
defines a divergent series and hence, by dropping the first
terms we can assume without loss of generality that S
1
> 1.
• Part (1). We have to prove that
P
a
k
S
k
is divergent. This is a particular case of
Theorem 1 with f(x) =
1
x
. We include here the proof I gave to the students.
Let us start by observing that
log(S
K
) = log
K
Y
k=1
S
k
S
k−1
!
=
K
X
k=1
log
S
k
S
k−1
= −
K
X
k=1
log
S
k−1
S
k
= −
K
X
k=1
log
1 −
1 −
S
k−1
S
k
= −
K
X
k=1
log
1 −
a
k
S
k
.(1)
If a
k
6= o(S
k
), the result is trivial. Hence, we assume lim
k→∞
a
k
S
k
= 0, and so, for
k > K
0
, 0 <
a
k
S
k
<
1
2
. Then, the inequality
x < − log (1 − x) < 2x (2)
valid for any 0 < x <
1
2
, gives us for any K > K
0
in (1),
log(S
K
) < −
K
0
X
k=1
log
1 −
a
k
S
k
+ 2
K
X
j=K
0
a
k
S
k
,
and the result follows.
• Part (2). Since
P
a
k
S
k
(log(S
k
+1))
2
<
P
a
k
S
k
(log(S
k
))
2
, it is enough to prove conver-
gence of the second series. Now, by (2),
a
k
S
k
< − log
S
k−1
S
k
=
Z
S
k
S
k−1
1
t
dt
4 Jorge Jiménez Urroz Normat 1/2012
and so, summing for k > 1,
X
k≤K
a
k
S
k
(log(S
k
))
2
<
X
k≤K
Z
S
k
S
k−1
1
t log(t)
2
dt =
Z
S
K
S
1
1
t log(t)
2
dt < +∞
The result follows.
3 Examples
Corollary 3
P
k≤K
1
k
diverges
Proof: Trivial from Theorem 2 and the divergence of
P
k≤K
1.
Corollary 4 The series
P
n
n
n
n!e
n
is divergent.
Proof: For any n ≥ 1, the inequality
e < (1 +
1
n
)
n+1
,
follows from (2) with x =
1
n+1
. Hence
(n+1)
n+2
(n+1)!e
n+1
>
n
n+1
n!e
n
> · · · ≥
1
e
, and so the
series
P
n
n
n+1
n!e
n
is divergent. Moreover, S
n
=
P
n
j=1
j
j+1
j!e
j
>
n
e
, and so
X
n
n
n
n!e
n
>
1
e
X
n
n
n+1
n!e
n
S
n
.
The result now follows by Theorem 2.
Corollary 5 Let f
0
(t) ≥ 0 a decreasing function, and f(0) > 0. Then,
P
f
0
(n)
diverges if and only if
P
f
0
(n)
f(n)
diverges. Moreover,
P
f
0
(n)
f(n)(log(f(n)+1))
2
always con-
verges.
Proof: Again, in the case when
P
f
0
(n) is convergent, both results are trivial by
noting that f(n) ≥ f(0), so we will assume
P
n≤N
f
0
(n) → ∞ with N . Let us prove
the first part of Corollary 5. Now, since
S
n
=
X
1≤j≤n
f
0
(j) <
Z
n
0
f
0
(t)dt = f(n) − f (0) < f(n), (3)
we deduce that f(n) → ∞ with n. Moreover,
S
n
>
Z
n
1
f
0
(t)dt = f(n) − f (1) >
1
2
f(n),
Normat 1/2012 Jorge Jiménez Urroz 5
for n sufficiently large. Hence,
X
n≤N
f
0
(n)
S
n
< 2
X
n≤N
f
0
(n)
f(n)
,
and the result follows from Theorem 2.
The second part of Corollary 5 follows from the second part of Theorem 2 and
(3).
Let log
1
(x) = log x, and for any integer j, log
j+1
(x) = log(log
j
(x)).
Corollary 6 For any integer J,
P
k
1
k
Q
j≤J
log
j
k
is divergent. On the other hand
P
k
1
k
Q
j≤J
log
j
k(log
J+1
k)
2
is convergent.
Proof: In Corollary 5, take f
J
(t) = log
J
(t). Note that for any f
J
(t) = log f
J−1
(t)
we have f
0
J
(t) =
f
0
J−1
(t)
f
J−1
(t)
. Now, Since f
0
J
(t) =
1
t
Q
j≤J−1
log
j
t
=
f
0
J−1
(t)
f
J−1
(t)
, we just have
to use Corollary 3, Corollary 5, and apply induction. For the second part, use the
second part of Corollary 5, (note that for any J and t sufficiently large depending
on J, log
J
t >
1
2
log
J
(t + 1)).
We include one final example just to show the wide range of applications of
this criterion. We will use it to give a new proof of a well known fact in Analysis,
consequence of (L
2
)
∗
= L
2
.
Corollary 7 Let (X, µ) an space of measure with µ(X) < ∞. Suppose f : X → R
is a measurable function such that
Z
X
|fg|dµ < ∞,
for any g ∈ L
2
(X). Then, f ∈ L
2
(X).
Proof: Without lost of generality we can assume f ≥ 0. By taking g = 1 we see
that f ∈ L
1
(X). Let us call A
k
= {x ∈ X : k ≤ f (x) < k + 1}. Then
X
k≥0
kµ(A
k
) ≤
Z
X
fdµ < ∞. (4)
Now suppose f 6∈ L
2
(X). Then
X
k≥0
(k + 1)
2
µ(A
k
) >
Z
X
f
2
dµ = ∞,
and so, by (4)
X
k≥0
k
2
µ(A
k
) = ∞.
6 Jorge Jiménez Urroz Normat 1/2012
Now, let us call S
k
=
P
k
j=1
j
2
µ(A
j
), and consider g(x) =
k
S
k
for any x ∈ A
k
. Then
g ∈ L
2
(X) since
Z
X
g
2
dµ =
X
k≥0
k
2
S
2
k
µ(A
k
) < ∞
by the second part of Theorem 2, (note that S
k
> (log(S
k
+ 1))
2
for k sufficiently
large), meanwhile
Z
X
fgdµ >
X
k≥0
k
2
S
k
µ(A
k
) = ∞,
by the first part of Theorem 2. Hence, we get a contradiction and the result follows.
Clearly both, Theorem 2 and Corollary 5, seem to have a wide variety of appli-
cations, and we leave to the interested reader to find new ones.
Acknowledgment: This note started when I was trying to prove, in an elementary
way, Corollary 7. I would like to thank J. L. Varona for pointing out the reference
(4) in which (1), and (3) are mentioned. I also want to thank Santiago Egido for
sharing with me the first example described at the beginning of Section 2 and
F. Chamizo for his corrections. Finally, and specially, I want to thank the referee
which pointed out the excellent book by De la Vallée Poussin (7), a gem, as well
as other suggestions.
Referenser
[1] N. H. Abel, Crelle, Vol 3, p. 81, 1828.
[2] B. Demidovich, Problemas y Ejercicios de Análisis Matemático, Paraninfo,
1978.
[3] U. Dini, Sulle serie a termini positivi, Anali Univ. Toscana, Vol. 9, 1867.
[4] K. Knopp, Theory and application of infinite series, Dover, 1990.
[5] M. Spivak, Calculus, Cambridge University Press, 2006.
[6] W. Rudin, Análisis real y complejo, Mc Graw-Hill, 1988.
[7] Ch.-J. de la Vallée Poussin, Cours d’Analyse infinitŐsimale. Louvain: Librairie
