Help required to sum an infinite series in a given equation

[chiraganand]

Hi,

I have a particular equation in a paper, wherein the author specifies an infinite series. The author has apparently found the sum of the series and calculated the equation. Can anyone please help me in understanding how to sum such a series. I have attached part of the paper with the equation and the the variables.

Thanks and Regards!

 

[PeroK]

I didn't see what answer you think he got for the sum. He just leaves it as a sum as far as I can see.

 

[chiraganand]

I didn't see what answer you think he got for the sum. He just leaves it as a sum as far as I can see.

The author doesn't actually solve it. he leaves it like this and then shows the end modeling results in a graph. So i am assuming, that he ended up solving it somehow or else he would have mentioned the number of terms he considered

 

[Antarres]

The author doesn't actually solve it. he leaves it like this and then shows the end modeling results in a graph. So i am assuming, that he ended up solving it somehow or else he would have mentioned the number of terms he considered

Well, that depends on the range of variables he had shown in the graph. He can choose just enough terms to have no visible deviation between the sum of the series and the graph he is showing with finite terms, depending on the space that the graph is covering. There is not enough information in this attachment to infer how such a series can be summed, in my opinion, and just because the graph is shown doesn't mean that the series had been summed. It may even be divergent(can't see if it is or isn't since I don't know $A_n$, $B_n$ and values of various parameters in there), but finite terms could approximate the value of some function he is looking for sufficiently good even if series starts diverging at some point. These types of asymptotic series are not uncommon, so I can only conclude that I don't have enough information to give you the definite answer, and that you don't need to assume that the series can be summed based on a graph(especially if it's a graph and not analytic form) that is shown to you.

 

[chiraganand]

Well, that depends on the range of variables he had shown in the graph. He can choose just enough terms to have no visible deviation between the sum of the series and the graph he is showing with finite terms, depending on the space that the graph is covering. There is not enough information in this attachment to infer how such a series can be summed, in my opinion, and just because the graph is shown doesn't mean that the series had been summed. It may even be divergent(can't see if it is or isn't since I don't know $A_n$, $B_n$ and values of various parameters in there), but finite terms could approximate the value of some function he is looking for sufficiently good even if series starts diverging at some point. These types of asymptotic series are not uncommon, so I can only conclude that I don't have enough information to give you the definite answer, and that you don't need to assume that the series can be summed based on a graph(especially if it's a graph and not analytic form) that is shown to you.

Thank you for your reply. Ok so I have the values of the various parameters are given below. Also is there a way to find the partial sum which converges? I have tried a number of values and I do know that after some terms the answer remains constant, but this also depends on d etc, hence to emperically find the number of terms is quite difficult.
d=3 mm, zn = (k0*a^2/2Bn), kp0=8485, kp1=4742, a=5 mm,r=0, r2=0.93, z=32 mm, eta=kp0/kp1
A=11.428 + 0.95175i
0.06002- 0.08013i
-4.2743 - 8.5562i
1.6576 + 2.7015i
-5.0418+ 3.2488i
1.1227 - 0.68854i
-1.0106 - 0.26955i
-2.5974 + 3.2202i
-0.1484- 0.31193i
-0.2085 - 0.23851i
B=
4.0697 + 0.22726i
1.1531 -20.933i
4.4608 + 5.1268i
4.3521 +14.997i
4.5443 +10.003i
3.8478 +20.0779i
2.528 -10.31i
3.3197 - 4.8008i
1.9002 -15.82i
2.634 +25.009i

 

[Antarres]

Hmm, well I don't think I am able to sum it, it looks quite comlicated. Partial sums are always finite, it just depends whether they converge or not. Looking at the form of $z_n$ it looks like a sum of Gaussians modified in a certain way, so looks logical that it will converge somehow, you can try applying convergence tests to see how it works. However, it's not really important whether it converges or not, it's just important to which term the series is terminated to approximate the result correctly. And that just depends on precision with which you want the result. If this is a theoretical formula he got(maybe it's empirical I wouldn't know), then he probably figured out whether it converges or not. You're just interested in the graph with some precision by summing some finite number of terms. Judging by its form, it's also likely that it converges with a good rate, that is, you don't need too many terms to form some viable result.

Those are just things I can guess by inspecting it, I don't have means to completely sum it to give you deeper information, so I'm just talking how such results are usually treated. If someone knows better, maybe they can help you with it.

 

[chiraganand]

Hi,

thank you for the reply. Hmm so bascially its the author who knows about it then. Also if i wanted to do a convergence study how would I go about it?

 

[Antarres]

Well you would have to simplify the form of the general term in the series as much as you could, and do some comparison tests for example. Now it would be good if you had formula for $A_n$ and $B_n$ in terms of $n$, so then you would find the actual form of the series. In case you don't, you can find some boundaries for those two parameters based on values you have and then work with comparison tests. It looks like a complicated task, but you can try it out.
What I said earlier was that delving into such calculations usually reveals nothing too important, so it depends on the purpose you need the series for. The series could've just be computed term by term until it was found that it is sufficiently converging, or something similar, if it was to be used for something practical. Theoretically, it would probably be quite involved to analyze it, but I'm not sure such analysis would give any better understanding, than the one given before when all the parameters and functions in the series were introduced.
That said, you have various tests for convergence, like integral test, quotient test, Raabe test etc. But every test pretty much stems from some form of comparison test, so that would be the most general way to do it, finding some inequalities that would lead you to check whether it's convergent or not.