The Method of Differences

Au101 · 2011-05-23 06:09:25

Hi guys,

I have a problem with the method of differences, in relation to series. I am given to believe that the method relies upon terms cancelling with the preceding or succeeding term and it is introduced in my textbook thus:

Which implies that the method only applies for series of this form - however, most of the questions which I have seem to involved series which are not, in fact, of this form. My first question, then, is basically if somebody could explain this method more thoroughly - preferably starting from the basics, but working-up to a rigorous 'professional' definition. The majority of my confusion, however, has arisen from my being unsure how to lay out answers to these questions, but I think it might be better to post specific question examples later.

Thanks

Last edited by Au101 (2011-05-23 06:09:43)

bobbym · 2011-05-23 06:20:36

Hi;

Best to see this work in practice. You already have the definition up there.

Supposing you needed to sum this;

Au101 · 2011-05-23 06:31:42

Ooooh thanks for yet another swift reply bobbym. That's really nice, I understand that perfectly, but I am rather confused about questions like this:

I was able to show part (a) fine, but I struggle with part (b) because I don't believe that this expression can be put in the form:

Once more - thank you very much

Last edited by Au101 (2011-05-23 06:32:20)

bobbym · 2011-05-23 06:41:50

Hi;

Although that is a convoluted mess they did there, the problem is solved for you. They have already split it into a difference of two functions.

Au101 · 2011-05-23 06:47:04

Hi, thanks bobbym.

Okay, I understand that, but I think this is what's confusing me - I felt that, since those two functions aren't of the form:

I was confused about my original definition of the method of differences, surely this needs to be refined? Apart from that, I think I see how your layout could be used, though, so that's one problem solved:). The layout which the book uses is very convoluted and - I don't mean to appear snobbish - but somehow ugly and messy and I quite like mathematical reasoning and I like to lay-out my solutions properly, but the alternative method which I found seemed to rely upon proving that relation, which isn't applicable here.

bobbym · 2011-05-23 07:02:15

Hi Au101;

There is a difference between authors and practitioners. Sometimes they are the same person but mostly not. Usually they are guys who read big books and write smaller books.

Au101 · 2011-05-23 07:08:50

Hehe A very good point, one which I'm becoming acutely aware of, because I am trying, myself, to put together a few notes. Not like a real textbook for publishing, or anything, but I've essentially been writing my own textbook so that when I forget all of this in about two years or something, I have something to look back on. And, I suppose, the first thing I want to do is to get a good explanation of the method of differences, because the whole

Definition doesn't seem to quite fit the bill and I'm not sure how else I can define the method, before I worry about trying specific examples.

Last edited by Au101 (2011-05-23 07:09:17)

bobbym · 2011-05-23 07:14:03

This is what I believe is happening.

I do not use that definition much and neither does anyone I know off. We say such a definition is formal, fancy talk for, of no practical use. Generally what you do is try to split it into differences, expand and cancel out terms. Like I have done above. This method is algebraic, easy to understand and you will never forget it.

Au101 · 2011-05-23 07:23:45

Oh, okay I think I'm happier with it now. But I am now slightly more concerned. In this question its easy, but what about in instances where the differences aren't given and I have to derive them myself. If the form f(r) - f(r + 1) is flexible, then how do I know how to split my series up so that the differences cancel like that.

bobbym · 2011-05-23 07:29:49

That is an answer that requires experience. You will have to see a few and do a few more. Submit ones that you are having problems with, I will try my best.

Au101 · 2011-05-23 07:36:25

Oh I see, so it's intuition and recognition, over a hard-and-fast method. Well, I guess I feel a bit better now, it was just that the book confused me somewhat, by proposing something of a standard form and then just completely ignoring it.

Thank you ever so much, yet again:)

bobbym · 2011-05-23 07:44:58

Hi;

There are some methods. Particularly the method of partial fractions. This can be a chore in itself.

Oh I see, so it's intuition and recognition,

All math is like that. Mathematicians will not admit it but mathematics is an art. If it were not we would all be like Euler!

The summation calculus is generally considered more difficult than integral calculus. Remember it is not the only tool we use in the summation calculus. There are other tools and between them you will get the job done.

Forman S. Acton probably the greatest numerical analyst wrote a book called "Numerical Methods That Work." We try to learn methods, not formal definitions that shed no light on the problem. We learn methods that get answers. Someday all math books will be of that type.

Math Is Fun Forum

#1 2011-05-23 06:09:25

The Method of Differences

#2 2011-05-23 06:20:36

Re: The Method of Differences

#3 2011-05-23 06:31:42

Re: The Method of Differences

#4 2011-05-23 06:41:50

Re: The Method of Differences

#5 2011-05-23 06:47:04

Re: The Method of Differences

#6 2011-05-23 07:02:15

Re: The Method of Differences

#7 2011-05-23 07:08:50

Re: The Method of Differences

#8 2011-05-23 07:14:03

Re: The Method of Differences

#9 2011-05-23 07:23:45

Re: The Method of Differences

#10 2011-05-23 07:29:49

Re: The Method of Differences

#11 2011-05-23 07:36:25

Re: The Method of Differences

#12 2011-05-23 07:44:58

Re: The Method of Differences

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