Further Formulae for Harmonic Series with Convergence Rate “−1/4”

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Title: Overview of the Closed Formulae for Infinite Series of Convergence Rate “−1/4” with Harmonic Numbers

1. The topic is mathematically relevant and specialized, offering insights into slow-converging harmonic series.
2. Derivations seem correct but would benefit from clearer explanations and added justification for some formulas.
3. The manuscript lacks clarity in key transitions and definitions. Equation formatting and better structure are recommended.
4. The paper needs a stronger connection to existing literature on harmonic series and convergence analysis.
5. No numerical examples or plots are included. Adding these would enhance understanding of the convergence behavior.
6. Define convergence rate clearly, add examples, label equations, and strengthen literature context.

Comments on the Quality of English Language

Title: Overview of the Closed Formulae for Infinite Series of Convergence Rate “−1/4” with Harmonic Numbers

1. The topic is mathematically relevant and specialized, offering insights into slow-converging harmonic series.
2. Derivations seem correct but would benefit from clearer explanations and added justification for some formulas.
3. The manuscript lacks clarity in key transitions and definitions. Equation formatting and better structure are recommended.
4. The paper needs a stronger connection to existing literature on harmonic series and convergence analysis.
5. No numerical examples or plots are included. Adding these would enhance understanding of the convergence behavior.
6. Define convergence rate clearly, add examples, label equations, and strengthen literature context.

Author Response

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Reviewer 2 Report

Comments and Suggestions for Authors

In the peer-reviewed paper, the authors discuss the closed formulae for series of convergence rate “-1/4” with harmonic numbers. This is an interesting and well-structured study.

The results obtained by the authors generalize and complement the relevant previously known ones. They all are carefully stated; details are given.

The paper provides relevant references so that readers can verify the correctness of the conclusions drawn.

However, the paper does not conclude applications motivated by concrete real-life problems, where the results apply and the hypotheses are verified. Providing relevant examples would have made the paper much more interesting with not only the theoretic value.

In addition, it would be good if the authors gave a geometric interpretation of the convergence of partial sums of series with harmonic numbers to constants. It is also worth discussing the results of the paper in the context of their applications (theoretical and/or practical) and outlining possible further directions of research.

Finally, since the results were obtained using computer algebra, more information about the software used should be added (e.g., in the appendix).

Author Response

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Reviewer 3 Report

Comments and Suggestions for Authors

In this manuscript the authors devise various sums involving harmonic numbers. Most of the sums are in fact some special values of hypergeometric functions or can be reduced to such values. My main concern is that the whole paper seems to be organized as a pile of identities and it is not clear what its main focus is. All the identities presented are obtained from some general formulas (which have been proved elsewhere and rightfully cited), by playing with various choices of coefficients $a,b,c,d,e$... The only original proofs given in the manuscript are those of the identities (24)-(30), but these proofs tend to be straightforward.

While some of the results may be interesting, I am personally against utilizing too many unnecessary notations, because it makes reading less comfortable. This includes symbols introduced on page 2 (line 28), which do not really reduce lengthy expressions; also the $O_n$, which are easily expressed in terms of the $H_n$. In equation (7), what is $V_n$ (perhaps $\Delta_n$)? I am not sure what the authors mean by the phrase "well-poised". The notation $\Omega_m$ is abused, first for the Bell polynomials (4), then redefined on page 5 (l.75).

All in all, I think that at this time the paper does not contain enough new material for publication. However, I am not definitely rejecting it. Adding some new and more general identities of the type of (24)-(30) may help. Therefore, I propose a major revision.

Author Response

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Reviewer 4 Report

Comments and Suggestions for Authors

Summary: The article’s topic is interesting. The article systematically applies the “coefficient extraction method” to the symmetric transformation of hypergeometric series by Chu and Zhang (2014). It explores a broad class of infinite series with convergence rate “−1/4,” linked to harmonic numbers. The paper establishes several closed-form expressions for these series in terms of mathematical constants such as π, ln 2, and the Riemann zeta values, serving as a valuable reference for future work.

 

• There is no specific details about “coefficient extraction method” and how it is a powerful method than any other method.
• All the results presented here are verified through symbolic computations. Some of the closed forms can be verified directly or are well-known, but not all of them. There is no analytical proof of the obtained results, which is a requirement of mathematical findings. So, I suggest adding at least one analytical proof.
• The author mentions that they use symbolic computation using Mathematica (11). However, there is no evidence of code or use of any specific package. A details method of the use of software is required.

Author Response

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Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

While I am not fully satisfied how the authors addressed my questions, the manuscript has improved enough to be published.

Author Response

Thanks to Reviewer~3 for approving this revised version.

Reviewer 4 Report

Comments and Suggestions for Authors

1. Although the authors added a few references about coefficient extraction, the underlying question remains unanswered. The author is requested here to define the term "coefficient extraction".
2.  The analytic proof is not sufficient yet, but it is acceptable. 
3. Based on the author's response, I would like to suggest revising lines 63-64 and remove the terms like computer algebra and symbolic calculus, Rather simply, they use inbuild Mathematica code to derive and validate several closed form, otherwise authors should provide sample of code before acceptance. 

Author Response

Thanks to Reviewer~4 for the further comments. These modifications

demanded in the report of 2nd round are faithfully accommodated.