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Hi,
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]
?
Could you please tell me how to expand the following binomial expression?
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
I cant help but wonder what you intended to do with this equation, what you wanted to use it for.
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.
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.
Expanding the binomial in this way doesnt seem to do any good,...
i mean accomplish any purpose other than to show how an expansion might be done,
which doesnt apply to this thread even though it has interesting side lines
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.
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.
But going to more terms doesnt seem to help either with the problem with the selection of x.
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.
This could be due to numerical instability in the normal computer floating point unit which is limited in precision but i havent investigated this.
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.
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.
It wasn't simple to me but now I have looked it up carefully. I believe you have used this formula:
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
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.
**broken link removed**
@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