Turn on thread page Beta
    • Thread Starter
    Offline

    13
    ReputationRep:
    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...
    • Community Assistant
    Offline

    17
    ReputationRep:
    Community Assistant
    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?
    Offline

    16
    ReputationRep:
    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.
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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)
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by Radioactivedecay)
    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 )
    • Community Assistant
    Offline

    17
    ReputationRep:
    Community Assistant
    (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).
    • Thread Starter
    Offline

    13
    ReputationRep:
    (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!!!
    Offline

    16
    ReputationRep:
    (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 )
    • Community Assistant
    Offline

    17
    ReputationRep:
    Community Assistant
    (Original post by BubbleBabby)
    Thank you!!!
    No problem!
    • Thread Starter
    Offline

    13
    ReputationRep:
    (Original post by radioactivedecay)
    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!
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: February 23, 2018

University open days

  1. University of Bradford
    University-wide Postgraduate
    Wed, 25 Jul '18
  2. University of Buckingham
    Psychology Taster Tutorial Undergraduate
    Wed, 25 Jul '18
  3. Bournemouth University
    Clearing Campus Visit Undergraduate
    Wed, 1 Aug '18
Poll
How are you feeling in the run-up to Results Day 2018?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.