Cite: ... the difference of two series ... if one converges, then so does the other converge, but if one diverges, then so does the other diverge.

Hence, you are safe ... as long as S2 has convergence theoretically rather than just computationally.

According to:

https://www.quora.com/If-the-difference-of-two-series-converges-do-the-two-series

If S2 is a convergent series and C is a known transcendental number, then the series S1 = C + S2 is also convergent. This follows from the fact that if a series is convergent, then the sum of that series with a constant is also convergent.

To prove this rigorously, you could use the convergence tests for series. For example, if S2 is a positive series and C is a positive constant, then you could use the comparison test to show that S1 is also convergent. If S2 is a series that satisfies the absolute convergence test, then S1 also satisfies the absolute convergence test and is therefore convergent.

There are several other convergence tests that you could use to prove the convergence of S1, depending on the specific properties of the series S2 and the constant C. In any case, the key idea is that the convergence of S2 implies the convergence of S1 = C + S2, as long as C is a known constant (in this case, a known transcendental number).

Mohamad Dzul Syakimin Mohd Noor many thanks a lot. Now I have a special case where I have defined an equivalent system (I mean, a series) which has a logarithmic structure elevated at the power (1/n^3). I mean, my series works as a {log(f(n))}^(1/n^3) from n=1 to n= Infinite. Numerically the series seems to converge to a real value but I am looking for a criterio when the terms of the series are defined by such {log(f(n))}^(1/n^3) from n=1 to n= Infinite. Numerically the series should arrive to a what a criterion of analytical convergence would prove for such logarithmic structure elevated at the dumping term. I want to be agree that convergence numerically and the boundedness of the terms are pointing out that the series converges.

Many thanks

Mohamad Dzul Syakimin Mohd Noor I think that I have resolved my main doubt in my problem. It is well-known that if the terms of a series are positive (which is my case as my {log(f(n))}^(1/n^3) is positive) and the partial sums are bounded, something that is visible even in graphs too, then the series, I mean the sum of {log(f(n))}^(1/n^3) from n=1 to n=infinite is convergent. that explains why the initial series which is not the same definition of the current cannot be easily studied. Then, a second series which represents an important status of convergence of the first studied speaks about the convergence of all the problem. Perhaps it sounds not clear now but I hope to present my own problem about the series I am analyzing and to pointing out that one can represent an apparent divergent series by two or more convergent series according to the nature of the problem and to return to the initial problem and solve it. This thread has been fascinating . Many thanks

even if your function f(n) is something like 1/n^2

then your series diverges according to

https://www.wolframalpha.com/input?i=sum+%28ln%281%2Fn%5E2%29%29%5E%281%2Fn%5E3%29%2C+n%3D1..infinity

or if f(n) = 1/n^20 then divergence is still there

https://www.wolframalpha.com/input?i=sum+%28ln%281%2Fn%5E20%29%29%5E%281%2Fn%5E3%29%2C+n%3D1..infinity

Steftcho P. Dokov and Mohamad Dzul Syakimin Mohd Noor

The next step I am considering is to transform my series into the Abel summation formula's context because a series could be represented by that formula. However, the evaluation of the series seems to be easy term by term if we expand it and the plot of the function f(n) that represents the terms into the infinite series Sum(f(n)) from n=1 to n=Inf. f(n) seems to be bounded, I mean, the plot shows a lot of mini curves descending in such way that in the infinite the peaks are vanishing ( I mean the areas under the f(n), which are always positive areas) vanished in the infinite accomplishing the boundedness of an integral criteria and the positive terms f(n) never negative. As I have read, if the terms or addends f(n) of a series are positive and the boundedness of the area under the f(n) curve is noticeably achieved, these should prove that the series Sum (f(n)) from n=1 to n=inf is convergent (which seems to be happening when adding more and more terms of expansion of n). The thing is that the curves are several chain of mountains reducing its area until reaching a final zero area in the infinite, this process is slowly and achieved in the infinite (area of integral zero per partial sum in the infinite). My idea is to use Abel summation formula to establish an integral representation of my series as boundedness is seen.

Well, many thanks for the contributions provided it is a very important matter on series in my opinion.

Sounds great, good luck.

It is good that you have observed that the terms of the series are positive and the area under the curve of the function representing the terms appears to be bounded, as these are important criteria for determining convergence. However, it is important to note that the Abel summation formula is typically used to sum divergent series, rather than to determine convergence. To determine convergence, it is usually sufficient to use the criteria mentioned above (positivity of terms and boundedness of the area under the curve). It is also worth considering other convergence tests, such as the ratio test or the root test, which can provide additional insights into the convergence properties of the series.

Good luck.

Mohamad Dzul Syakimin Mohd Noor yes, I am convinced about that boundedness and positive terms, but for now, it is graphically I can observe the behavior when I evaluate the function f(n). The observation shows that I have a set of "hills" or "quasi-peaks" similar to a chain of hills vanishing in the infinite when growing n and are positive. I mean, the area under these mini hills is reaching zero evidently but I expected to find a formula or theorem to show it in analytical way