Hello MrAlHi,
Do you mean you want:
[LATEX]\sqrt{y^6+3 x^2 y^4+3 x^4 y^2+x^6}[/LATEX]
or
[LATEX](x^2+y^2)\sqrt{x^2+y^2}[/LATEX]
?
The expansion of a binomial expression is a straight forward operation. Why are you having trouble with it? Because the exponent is not a positive integer, the expansion will be an infinite series. In the attachment is the first six terms of the infinite series. Included is a comparison of the exact value and the first six terms of the binomial expansion.Could you please tell me how to expand the following binomial expression?
http://www.electro-tech-online.com/mathematics-physics/mathematics-physics/a...1&d=1346407856
PG1995,
The expansion of a binomial expression is a straight forward operation. Why are you having trouble with it? Because the exponent is not a positive integer, the expansion will be an infinite series. In the attachment is the first six terms of the infinite series. Included is a comparison of the exact value and the first six terms of the binomial expansion.
Ratch
By equation, I assume you mean the binomial expansion? I don't have any use for it, except to illustrate to PG how to expand a binomial expression.I cant help but wonder what you intended to do with this equation, what you wanted to use it for.
The full expansion is a infinite series, so the last terms will complete the symmetry.It's also pretty misleading to show solutions that match for a well chosen set of x and y alone. The equation isnt even symmetrical in x and y as the original surely is.
What you say is true, because the values you chose make the series diverge. Now, if you chose x=2, y=1, the radicand will not change, and the series will converge. So you want to chose x to be as large as possible compared to y to make the series converge faster.For example, chose x=1 and y=2 and the results vary from the original expression by some 400 percent so they are not even remotely close.
It showed PG how to expand a binomial. Its uses include calculating an approximate value.Expanding the binomial in this way doesnt seem to do any good,...
I don't know how to expand a binomial any other way.i mean accomplish any purpose other than to show how an expansion might be done,
It answered PG's question.which doesnt apply to this thread even though it has interesting side lines
A binomial expansion is what it is. You have to live with it.With an expansion i dont want to be forced to carefully choose my x and y, there are better expansions that are not so picky.
You will get convergence anytime x > y in (x+y)^n .It makes sense that any expansion should definitely include x=1 as it seems very silly to not be able to choose that value or close to that value.
The selection of x or y deals with convergence, not the number of terms.But going to more terms doesnt seem to help either with the problem with the selection of x.
If convergent, a greater number of terms bring the approximation closer to the true value.I am thinking that there may be a limit that has to be imposed (you might check into this), but it may actually get worse with more terms.
How you calculate high precision is up to you.This could be due to numerical instability in the normal computer floating point unit which is limited in precision but i havent investigated this.
Thank you. You can Google for the proof. Like I said, how you do high precision is up to you.But in showing the expansion i think you did a good job of illustrating that. What would be interesting would be to see a proof of the binomial expansion with non integer powers. But then again the practicality may be severely limited do to the aforementioned CPU floating point limitations even in the modern computer, and high precision numerical routines will slow things down quite a bit.
There are a lot of ways to do the same thing.I can think of one expansion that works pretty nice for square root. It involves taking the natural log however (but only once no matter how many terms in the expansion), but that can be achieved with another expansion. It's very stable over a very wide range of x too that goes right down to 0^+, the right side of zero.
Yes, I did.It wasn't simple to me but now I have looked it up carefully. I believe you have used this formula:
http://www.electro-tech-online.com/a...1&d=1346878426
Thank you, Ratch.
It wasn't simple to me but now I have looked it up carefully. I believe you have used this formula:
Best wishes
PG
The way I have written the series x=a and y=x. Yes, "a" should be as large as possible compared to x to make the series converge faster. For instance, in the highlighted row, there is a difference of 10,000 between the exact value and approximated value.Ratchit said:So you want to chose x to be as large as possible compared to y to make the series converge faster.
Thank you, Ratch, MrAl.
The way I have written the series x=a and y=x. Yes, "a" should be as large as possible compared to x to make the series converge faster. For instance, in the highlighted row, there is a difference of 10,000 between the exact value and approximated value.
@MrAl: I just saw your post. Thank you. Once I have read it carefully I will let you know if I have any queries. Thanks.
Regards
PG