binomial expansion C4

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#1
can someone explain to me what exactly limits are in binomial expansion and what they are used for? Can you also explain why you can use binomial expansion on an algebraic fraction but when you expand the fraction manually you get a different answer? Like the equation can be to the power of 2 but you could expand it to 100 if you wanted to and i don't understand why that is...
0
3 years ago
#2
What do you mean by the second question?

In C4, you deal with binomial expansions with non-natural N (powers) e.g. fractional or negative or both. These are expansions which go on forever (infinite) however these expansions are only value for certain x; (1+ax)^n expansion is only valid if |ax| < 1 --< |x| < 1/a. However you usually you only do it to a few powers as the later powers can become insignificant in estimations. I don't get your questions - can you clarify where you're confused and I'll try to help?
1
3 years ago
#3
When you do binomial expansion, you're essentially breaking up the sum to infinity of an infinte series into it's single terms. For instance 1/(x+1) is the sum to infintiy of the series(first term 1, multiplier is -x) 1-x+x^2-x^3....and so on. So when you use binomial you're essentially breaking thaat sum up and finding the separate terms of the series. However, binomial only gives you an approxiamtion because the series has infinitely many terms and you only use some some of them. The reason there is a limit for the value of x is because the fraction is undifined when the denominator equals zero, in the instance above, x<-1 or x>-1.
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#4
(Original post by thekidwhogames)
What do you mean by the second question?

In C4, you deal with binomial expansions with non-natural N (powers) e.g. fractional or negative or both. These are expansions which go on forever (infinite) however these expansions are only value for certain x; (1+ax)^n expansion is only valid if |ax| < 1 --< |x| < 1/a. However you usually you only do it to a few powers as the later powers can become insignificant in estimations. I don't get your questions - can you clarify where you're confused and I'll try to help?
i mean for example in this past paper https://madasmaths.com/archive/iygb_...apers/c4_d.pdf
in uestion 2 b there is an algebriac fraction. If you told me to expand it i would never get to that answer i would just expand my denominator . Also if i got the ewuation (x+3)^2 and i was told to expand this binomially to the power of 3 by the equation i would be able to do it however manually i would never get to the power of 3 hence my frustration. i just do not understand why if i was to expand the fraction manually and then through binomial why i would end up with different answers. 55555555 (crying face)
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#5
When you do binomial expansion, you're essentially breaking up the sum to infinity of an infinte series into it's single terms. For instance 1/(x+1) is the sum to infintiy of the series(first term 1, multiplier is -x) 1-x+x^2-x^3....and so on. So when you use binomial you're essentially breaking thaat sum up and finding the separate terms of the series. However, binomial only gives you an approxiamtion because the series has infinitely many terms and you only use some some of them. The reason there is a limit for the value of x is because the fraction is undifined when the denominator equals zero, in the instance above, x<-1 or x>-1.
What do you mean exactly by the sum to infinity ? ( sorry if this is a stupid question )
0
3 years ago
#6
(Original post by BubbleBabby)
i mean for example in this past paper https://madasmaths.com/archive/iygb_...apers/c4_d.pdf
in uestion 2 b there is an algebriac fraction. If you told me to expand it i would never get to that answer i would just expand my denominator . Also if i got the ewuation (x+3)^2 and i was told to expand this binomially to the power of 3 by the equation i would be able to do it however manually i would never get to the power of 3 hence my frustration. i just do not understand why if i was to expand the fraction manually and then through binomial why i would end up with different answers. 55555555 (crying face)
Okay so:

a) told you to find the partial fraction expression of the LHS
b) to expand it

So you expand each binomial:

A(1-x)^-1 + B(2-x)^-2 + C(2-x)^-1

You gotta put it into the forum (1+ax)^n so:

A(1-x)^-1 is fine

B(2-x)^-2 = B(2^-2)(1-x/2)^-2 = B/4 (1-x/2)^-2

C(2-x)^-1 = C(2^-1)(1-x/2)^-1 = C/2(1-x/2)^-1

Now, expand each three up to the powers of x^2 and collect terms. If x is small then any higher powers (cubes and so on) are insignificant and therefore if x is sufficiently small, that estimation (up to the squares) gives an accurate value of the LHS (original fraction).
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#7
(Original post by thekidwhogames)
Okay so:

a) told you to find the partial fraction expression of the LHS
b) to expand it

So you expand each binomial:

A(1-x)^-1 + B(2-x)^-2 + C(2-x)^-1

You gotta put it into the forum (1+ax)^n so:

A(1-x)^-1 is fine

B(2-x)^-2 = B(2^-2)(1-x/2)^-2 = B/4 (1-x/2)^-2

C(2-x)^-1 = C(2^-1)(1-x/2)^-1 = C/2(1-x/2)^-1

Now, expand each three up to the powers of x^2 and collect terms. If x is small then any higher powers (cubes and so on) are insignificant and therefore if x is sufficiently small, that estimation (up to the squares) gives an accurate value of the LHS (original fraction).
Thank you!!!
0
3 years ago
#8
(Original post by BubbleBabby)
What do you mean exactly by the sum to infinity ? ( sorry if this is a stupid question )
I meant the sum to infinity of an infinite geometric series.(sorry should've made it clearer). The fraction you start with in a binomial expansion question is in the form a/1-r , or some multiple of it, where a is the first term and r is the ratio you multiply each term with to get the next term. Binomial expansion helps you break this sum to infinity into regular terms.(hopefully my original post makes sense now )
0
3 years ago
#9
(Original post by BubbleBabby)
Thank you!!!
No problem!
0
#10
i meant the sum to infinity of an infinite geometric series.(sorry should've made it clearer). The fraction you start with in a binomial expansion question is in the form a/1-r , or some multiple of it, where a is the first term and r is the ratio you multiply each term with to get the next term. Binomial expansion helps you break this sum to infinity into regular terms.(hopefully my original post makes sense now )
thank you!
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