# C4 Binomial Expansion help

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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 :-)

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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(Original post by

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.

**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.

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?

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(Original post by

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 . This function has range solely . While it is true that both square to give 4, it is only true that . 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 , 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.

**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 . This function has range solely . While it is true that both square to give 4, it is only true that . 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 , 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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