# Power series expansion for Laplace transform

• Will
Originally posted by WillIn summary, the conversation was about finding the Taylor series expansion for the Laplace transform of e^t. After some back and forth discussion, it was concluded that the sum of a convergent series is equal to the first term divided by one minus the common ratio. There was also a debate about the best way to teach math and science, with one person suggesting a more step-by-step approach while the other argued for trial-and-error.
Will
[SOLVED] power series expansion for Laplace transform

We are to find the Taylor series about 0 of e^t, take the tranform of each term and sum if possible. So I know the expansion of e^t is 1+x/1!+x^2/2!... x^n/n! then taking the tranform, 1/s + (1/1!)(1!/s^2) +(1/2!)(2!/s3)... and so on then the factorials cancel and I have in summation notation sum(1,inf.)1/s^n. I can see that. But answer says that equals 1/(s-1) and I am not seeing that.

Originally posted by Will
sum(1,inf.)1/s^n. [...]that equals 1/(s-1) and I am not seeing that.

Hi Will,
that is similar to the Infinite Geometric Sum, which starts at n = 0 instead of n=1. You will probably find it in your lecture or textbook. Just modify that a little, and you have the proof you need.
If not, you can prove it directly. Just call the sum's value S, and compare terms like s*S, S+1, ...
Sorry I can't be more explicit. This is Homework Help, so I mustn't do it for you...

Originally posted by arcnets
This is Homework Help, so I mustn't do it for you...

I promise, this was not an assigned problem, but one of the odds that I was trying to do for practice. Anyway after reviewing my calculus text, I think I have something, please correct me if I am mistaken. The book says the sum of a convergent series is (1st term)/(1-common-ratio) so I have (1/s)/(1-(1/s)) = 1/( s-1) and this converges for 1/s < 1 or s > 1 ?? I am really rust on these infinite series things. I was pretty shaky in 2nd semester calculus, and we didnt use it at all un 3rd semester( now that was a fun class). Now I am in diff. eq. and I am hurting!

Is there an archive of worked examples you could refer me to? I think that its a shame that students who are copying problems and not wanting to learn them makes it harder for us that do. Because in my opinion, having the student spend hours trying to reinvent the solution of these problems is a waste of time for the most part. I ALWAYS try and do the problem on my own at first, but getting stumped really unerves me, I can't relax until I understand it and I can't even concentrate on other problems.
If I were a teacher, I would suggest problems, but give access to full solutions, but have rigorous tests with minimal notes. The students who weren't going through the problems, that would reflect on their preformance, plus there wouldn't be any credit for copied work. Because I have spent so many hours on problems, then when I see the solution, I was so close, it usually involved a relationship or formula that I had been stared at for hours, when if I had access to the solution, I would understand it quickly, and be able to to 5 or 10 more examples like it.
I think that the math and science should be tought like martial arts for example. The instructor shows you exactly what to do step by step. You don't try and go about it your own way until you have mastered what is known. Also people who don't put in the effort practicing don't get good. Anyway, sorry for the rambling post, just venting my academic frustration!

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Originally posted by Will
the sum of a convergent series is (1st term)/(1-common-ratio) so I have (1/s)/(1-(1/s)) = 1/( s-1) and this converges for 1/s < 1 or s > 1 ??
Looks OK to me.
Is there an archive of worked examples you could refer me
Sorry, no. There's a site called mathworld which is often referred to by members of this forum. Maybe someone else can help...
I think that the math and science should be tought like martial arts for example. The instructor shows you exactly what to do step by step.
Hmmm... If you teach like this, then maybe the students learn what to do. But will they also learn why they should do it like this? And will they develop their own personal way to (using your picture) 'fight'? I doubt it...
I strongly believe that trial-and-error is needed when you want to get the knack of math & science. I also believe you have to spend a lot of time doing this. Nor do I think that this time is wasted. One reason is, what we know now was largely discovered by trial-and-error in the first place. Many major discoveries were made when the researchers were looking for something completely different from what they finally found.
Think of science as a landscape you want to discover. Doing it like martial arts would mean, you race along the highway. But the more interesting sights may lie offroad in the countryside, only to be reached by small, untrodden & dangerous paths.

My experience is, it does good to do trial-and-error in a team. Because different minds work differently, and one will seee what the other does not immediately see. Saves time, too!

I can see you're frustrated. Well, probably everybody is, sometimes. Hope you get over it soon.

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Will

That was a good point about things being discovered by trial and error. That is a good argument for your point. I still disagree for several reasons:
First, in all of my classes, when someone gets stuck,in most cases there stuck good, until they can go to the teacher or another student and have him/her thoroughly explain things. So it seems to make more sense to have solutions readily available. I mean is a first a second DFQ student really ging to come up with some original theorem, or have the best ways to solve these problems been worked out long ago? Maybe I'm wrong becuase like you said all our math knowledge to date has come from trial and error. I just feel maybe that should be left to the really advanced people, and students should just learn the mechanics of doing problems,problems, and more problems, plus hard tests.
I also disagree about being less clear about the reasons behind why to do vs. what to do. A good explanation sinks that part in even more thorougly for me. Maybe that's because I truly want to learn, and like stated earlier, I ALWAYS try to do the problem on my own first, thus developing a little personal style, and when I do have to peek at them, I work the problem even more, and often throw my own twist and try to modify portions to understand the "why" part. Thats when it sinks in long term.
That was an exellent point about the trial and error by group. I think that is the pitfall of many science/math student, that they think they have to figure it all out on their own. Maybe that is why I am for open acess to solutions. The ones who maybe arent so social try and do it on their own, and when they get stuck they feel stupid.
I hope I didnt give you the wrong idea with the martial arts analogy. I meant most skills, being a musician, or even a surgeon. The don't tell a even a third or fourth year med student, "try and come up with the best way to transplant the liver, I can't tell you exacly how to do it"?! Just kidding! But seriously, what I am saying is the people making the discoveries are one the have learned the fundamentals solid. Is there really that much "creativity" at this level?
Anyway, sorry for yet another off topic rant, I have to get back to studying for finals. Thanks for your help, I will definately be back for more.

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Originally posted by Will
First, in all of my classes, when someone gets stuck,in most cases there stuck good,

Yes, I agree that's frustrating. I may sound cynical, but the step where you got stuck is probably the most (or only) important step in the whole proof. Most proofs have just one central creative idea, and the rest is technique.
(Just think of the 180°-rule for triangles. The central idea is that you can pave the plane with triangles of the same sort, regardless of what they look like. The rest is using basic definitions and simple algebra.)
So, if you get stuck, you have already learned something: that the next step is the important one.
I mean is a first a second DFQ student really ging to come up with some original theorem,
Definitely: NO! It's not fair to expect from a student that he invents theorems. But I think it's helpful if a teacher states problems that can be solved with the 'toolbox' of theorems and definitions from the lecture. The student's job is to fiddle about with the 'tools' until he can solve the problem. A good problem should be of such a kind that, after seeing the solution, the student slaps his head and says "Oh boy, this was so easy, I must have been blind!"
students should just learn the mechanics of doing problems,problems, and more problems, plus hard tests.
Agree, but you may call it a problem only if it has some basic difficulty, i.e. if anything is to be learned in solving it.
I think that is the pitfall of many science/math student, that they think they have to figure it all out on their own.
Totally agree. Team work is the method of our time. Just think of how many names appear on scientific publications!
The ones who maybe arent so social try and do it on their own, and when they get stuck they feel stupid.
Yes. There's still the image of mathematicians and scientist being sort of 'geeks', people who don't socialize easily, like wizards in an ivory tower. I think that's outdated.
"try and come up with the best way to transplant the liver,
Good point. You can't have patients dying all the time because students do trial-and-error. But isn't this exactly the reason why someone becomes a mathematician and not a medic? You don't only want to follow rules, you want to invent, play around and sometimes be surprised...
Is there really that much "creativity" at this level?
Yes! I'm convinced your creativity level is highest while you study. Maybe you will never be so creative again...

## 1. How is a power series expansion used to find the Laplace transform of a function?

A power series expansion is used to find the Laplace transform of a function by expressing the function as an infinite sum of terms, each with increasing powers of a variable. The expansion is then substituted into the Laplace transform integral and the integral is evaluated. This allows for the Laplace transform to be expressed as a series of terms, making it easier to solve for the transform of the original function.

## 2. What are the benefits of using a power series expansion for Laplace transform?

Using a power series expansion for Laplace transform allows for functions that cannot be directly transformed using the Laplace transform integral to be transformed. It also allows for the transformation of functions with complicated algebraic expressions or functions that are piecewise defined. In addition, the expansion can also provide a more precise approximation of the original function than other methods of finding the Laplace transform.

## 3. How does the number of terms in a power series expansion affect the accuracy of the Laplace transform?

The number of terms in a power series expansion directly affects the accuracy of the Laplace transform. The more terms included in the expansion, the more accurate the resulting Laplace transform will be. However, including too many terms can lead to a cumbersome calculation and may not be necessary for the desired level of accuracy.

## 4. Can all functions be transformed using a power series expansion?

No, not all functions can be transformed using a power series expansion. Functions must meet certain requirements, such as being continuous, to be able to use a power series expansion for Laplace transform. In addition, some functions may have a convergence radius that limits the accuracy of the expansion.

## 5. Are there any limitations to using a power series expansion for Laplace transform?

One limitation of using a power series expansion for Laplace transform is that it can be a time-consuming process, especially when a large number of terms are needed for accuracy. In addition, there may be cases where the expansion does not converge, making it impossible to use this method to find the Laplace transform of a function. It is important to carefully consider the complexity and convergence of the expansion before using this method.

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