1. Introduction
Usually there is no necessary connection between the convergence of series and infinite integrals [1] , but when the product function has certain characteristics, the two are in the same convergence state, for example, the infinite integral of a monotonically decreasing function has the same convergence as its corresponding series [2] . In the fourth edition of the eight cups competition, held on 1 August 2022, the eighth question of the mathematical group B appeared the problem of determining the convergence of a sinusoidal series: given the parameter
, try to discuss the convergence of the series
with respect to the value of the parameter p (when converging, it should be determined whether it is absolutely convergent). We find that the level when
, the level of dispersion; when
, the level of conditional convergence; when
, the level of absolute convergence, and the infinite integral
has the same convergence. It can be seen that the class of non-monotonic functions has the same convergence of the series and the corresponding infinite integral under certain conditions. The above shows that the sinusoidal series
has the same convergence as the infinite integral
when
. In the following, we try to extend this conclusion to the general
case, to begin with we study the convergence of the series
, then we discuss the convergence of the infinite integrals
, and by comparing the two, we conclude that the series
and the infinite integrals
have the same convergence in this paper.
2. Convergence States of Sinusoidal Term Levels
Proposition: For the sinusoidal series
(parameter
), the series diverges when
, converges conditionally when
, and converges absolutely when
.
Corollary: Let any real number
and parameter
be the convergence state of the sinusoidal series
: when
, the series diverges; when
, the series converges conditionally; when
, the series converges absolutely (provided that
).
Proof:
Using the sum and difference product formula, express
as
(1)
According to Taylor’s formula, it is possible to obtain
(2)
(3)
The association of (1), (2) and (3) yields
(4)
Therefore, when
is found
(5)
According to by the Cauchy criterion
does not exist, so the level
diverges, and by the A-D discriminant, the levels
all converge, and the p-levels also converge, so the level
diverges.
When
, the level
, it is easy to know that the p level converges, there is a comparative discriminant method to get the level
absolute convergence.
When
, by the A-D discriminant, the level
converges, the following consider the convergence of the level
, for
, there is
, it is not difficult to see that the p level
diverges, and
converges, so
diverges, so
conditional convergence.
When
, the level
is divergent, also use the inverse method to prove that the level is divergent. First of all, assume that there is a point
, so that the level of
convergence, to find out the level of n before the part of the term and
is bounded on n, according to the A-D method of discrimination can be
convergence, which is contradictory to the
divergence, so when
, the level of
divergence.
In summary, the sinusoidal series
(
,
), when
, the series diverges; when
, the series converges conditionally; when
, the series converges absolutely. (a)
Consider below the case where
takes values at the endpoints
and
:
When
is the level
,
is a constant, so it is in the same convergence state as the level p. That is, when is the level converges and is absolutely convergent; when is the level diverges. That is, when
, the level
converges and absolutely converges; when
, the level
diverges. (b)
When
, the convergence state of the level
is discussed below:
When
, the p-series converges due to
, and the series
converges absolutely by the comparative discriminant;
When
is used, by the product to sum formula, we know that
(6)
Using the A-D discriminant method,
part of the sum series is bounded, and
monotonically decreasing and tends to 0, so
convergence, the following proof of
conditional convergence, because
, and by the A-D discriminant method of the series
convergence, but the sum series
divergence, so
conditional convergence.
When
, by the A-D discriminant method, the part of the series
and the series is bounded, and
monotonically decreasing and tends to 0, so
convergence. And
, by the product and difference formula to get
(7)
Therefore, the part of
and the series are bounded, and
is monotonically decreasing and tends to 0, so the series
converges, but the series
diverges, so the series
converges conditionally.
In summary, when
, the level
When
, the level converges absolutely; when
, the level converges conditionally. (c)
Summing up at (a), (b) and (c), we have that the series diverges when
; the series converges conditionally when
; and the series converges absolutely when
. The corollary is proved. QED
3. Convergence States of Infinite Integrals
Lemma: Infinite integrals
, for any real number
, parameter
, at that time
, the infinite integrals diverge; at that time
, the infinite integrals converge conditionally; at that time
, the infinite integrals converge absolutely.
Proof.
Consider first the case of
. The infinite integral is transformed into
, and
is a positive constant, so the level is in the same convergence state as
. So when
, the series converges and converges absolutely; when
, the series diverges.
Next consider
. The infinite integral is transformed to
. When
is
, while
converges, by the comparative discriminant, we know that
converges and is absolutely convergent.
Consider
when
is bounded on
and
is monotone on
and
, so
converges. Since
, combined with the A-D discriminant, we know that
converges and
diverges. So
converges conditionally.
When
, the same process as above, consider that
is bounded on
and
is monotone on
and
, so
converges. But
, combined with the A-D discriminant, we know that
converges while
diverges. So
converges conditionally.
Finally, consider the case of
. Let
, then
,
, so that
(8)
When
is
,
,
, we know that
converges by the comparative discriminant, and thus
converges absolutely.
When
is
. On the one hand,
, has
, and
is monotonic and tends to 0 when
(
). It can be deduced from the fact that
converges according to the A-D discriminant. On the other hand, since
,
(9)
where
. According to the A-D discriminant condition, it is known that
is convergent and
is divergent (
), so the infinite integral
is divergent when
and thus
is conditionally convergent.
When
is
, substitution yields
(10)
It follows from the Cauchy convergence criterion that
,
and
, such that
(11)
Therefore
does not exist, so
diverges.
When
is
, the infinite integral
is divergent. Using the converse method, suppose the infinite integral
converges, then since
is bounded on
, according to the A-D discriminant, there should be
convergence, a contradiction. Therefore, when
,
is divergent.
To sum up: the infinite integral
, for any real number
, parameter
, when
, the infinite integral diverges; when
, the infinite integral converges conditionally; when
, the infinite integral converges absolutely. QED
4. Theorem on the Convergence State of the Sine Term Hierarchy with the Infinite Integral Homology
Theorem: Arbitrarily
, with parameter
, the level
is convergent to the same state as the infinite integral
.
Proof:
According to the above corollary, for a sinusoidal series
, any real number
, and the parameter
, the series diverges when
, the series converges conditionally when
, and the series converges absolutely when
.
According to the above lemma, for the infinite integral
, any real number
, parameter
, the infinite integral diverges when
, the infinite integral converges conditionally when
, and the infinite integral converges absolutely when
.
Accordingly, we obtain that the level
is homoconvergent with the infinite integral
(any
, parameter
). The proof of the theorem is thus complete. QED
5. Conclusion
Inspired by
when the sine series
and the infinite integral
(parameter
) are in the same convergent state, we explore the convergence of the series
and the infinite integral
when
is in the same convergent state, and we extend the conditions of the function class of the two in the same convergent state, expanding from monotonically decreasing functions to the class of non-monotonous functions, and we will continue to explore the other classes of the two in the same convergent state in the future.
Acknowledgements
In the first place, this paper is supported by the fund for the first-class undergraduate programme construction project (Mathematics and Applied Mathematics) of Xinjiang Uygur Autonomous Region in 2019, the fund for the first-class undergraduate programme construction project (Financial Mathematics) of Xinjiang Uygur Autonomous Region in 2021, the fund for the first-class undergraduate programme construction project (Probability Theory) of Changji College in 2022, and the project for the articulation of high schools and colleges (Quality Enhancement of Modern Vocational Education)-Mathematics and Applied Mathematics in 2022.
In the next place, the communication with Professor Liping Zhu has made the logic of proof, the line of thought, and the future progress of the work in this paper clearer and clearer.
Eventually, the authors are very grateful to the relevant literature for inspiring this paper and the journal reviewers for their valuable comments.
Project Funds
2019 Xinjiang Uygur Autonomous Region First-class Undergraduate Major Construction Project (Mathematics and Applied Mathematics) Fund.
2021 Xinjiang Uygur Autonomous Region First-class Undergraduate Major Construction Project (Financial Mathematics) Fund.
2022 Changi College First Class Undergraduate Curriculum Development Project (Probability Theory) Foundation.
Higher Education Bridging Project (Quality Enhancement in Modern Vocational Education)—2022 in Mathematics and Applied Mathematics.