hhattiecc
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Report Thread starter 6 years ago
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Would really appreciate any help with these parts of binomial expansion with -ve and noninteger exponents that I can't seem to get my head around (I've completely over thought it)

1. So when you have (1 - 10x) to the power of either a negative integer or fraction, I understand that you have to restrict the values of x so that the series converges, so |x|<1. So in the (1 - 10x) case |10x| < 1 and so |x|<1/10.
What I don't understand is that when you expand it to give (say you raise it to the power of 3/2) to give 1-15x + 75/2 x^2..., if you put any value in for x so long as |x|<1 it would still converge, despite the fact that some are bigger that 1/10 , because of the increasing exponent.

2. When you have something raised to the power of a fraction with an even denominator, e.g (4 + x)1/2, and then you take the 4 out to make 41/2(1 + x/4)1/2 why is it then presumed that 41/2​ takes the value of 2 rather than -2. I can't get my head around why it isn't ±2 and again when you do the expansion with the (1 + x/4), why wouldn't the expansion of that be ±?

3. I really don't understand how the binomial expansion can give an infinite expansion when if you just leave it with the exponent as a fraction/negative integer it will give an exact answer. So 1/(1 + x) for example. If x is 1/2 then the answer is exactly 2/3, but using binomial expansion when you put (1 + x) to the power of -1 gives you an infinite expansion that will never give you the answer of exactly 2/3, however close it may get, so why ever use the expansion when technically it's incorrect?

SO confused

Thank you :-)
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Smaug123
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Report 6 years ago
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(Original post by hhattiecc)
What I don't understand is that when you expand it to give (say you raise it to the power of 3/2) to give 1-15x + 75/2 x^2..., if you put any value in for x so long as |x|<1 it would still converge, despite the fact that some are bigger that 1/10 , because of the increasing exponent.
No, this doesn't converge unless |x| < 1/10. Try it and see - you'll find that the coefficients grow faster than the x^n shrinks.

2. When you have something raised to the power of a fraction with an even denominator, e.g (4 + x)1/2, and then you take the 4 out to make 41/2(1 + x/4)1/2 why is it then presumed that 41/2​ takes the value of 2 rather than -2. I can't get my head around why it isn't ±2 and again when you do the expansion with the (1 + x/4), why wouldn't the expansion of that be ±?
This is a misunderstanding of the function x \mapsto x^{\frac{1}{2}}. This function has range solely x \geq 0. While it is true that both \pm 2 square to give 4, it is only true that 4^{\frac{1}{2}} = 2. Functions return only one value.

3. I really don't understand how the binomial expansion can give an infinite expansion when if you just leave it with the exponent as a fraction/negative integer it will give an exact answer. So 1/(1 + x) for example. If x is 1/2 then the answer is exactly 2/3, but using binomial expansion when you put (1 + x) to the power of -1 gives you an infinite expansion that will never give you the answer of exactly 2/3, however close it may get, so why ever use the expansion when technically it's incorrect?
It's not incorrect. An infinite expansion will never give you the answer of exactly 2/3, of course, but that's only because we are imperfect and finite. The same question: why would you ever bother with using expansions of the number \pi, when technically any expansion of it is incorrect? Answer: because they're useful. I know that 3.14159 is close to pi (it is less than one part in ten thousand away from pi), and that's good enough for many applications.
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hhattiecc
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Report Thread starter 6 years ago
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(Original post by Smaug123)
No, this doesn't converge unless |x| < 1/10. Try it and see - you'll find that the coefficients grow faster than the x^n shrinks.


This is a misunderstanding of the function x \mapsto x^{\frac{1}{2}}. This function has range solely x \geq 0. While it is true that both \pm 2 square to give 4, it is only true that 4^{\frac{1}{2}} = 2. Functions return only one value.


It's not incorrect. An infinite expansion will never give you the answer of exactly 2/3, of course, but that's only because we are imperfect and finite. The same question: why would you ever bother with using expansions of the number \pi, when technically any expansion of it is incorrect? Answer: because they're useful. I know that 3.14159 is close to pi (it is less than one part in ten thousand away from pi), and that's good enough for many applications.
Thank you so much, reading back the 2nd question I had I'm slightly embarrassed as I really should have thought about that. Thanks again, the first question in particular was driving me mad.
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