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Questions about binomial expansions in C4 mathematics watch

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    Is it true that an expansion is never convergent if n is a positive integer ?
    Is the range of values for which an expansion is valid, the same as the range of values for which the same sequence is convergent (when n is less than 1 or is a positive non integer value)?
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    the reason we say that a the series is not convergent (it is divergent) when n>1 is because lets see the co efficient of x^3 in the expansion of (x+)^5;
    (n (n-1)(n-1))/3! ...if u try to replace any values of n>1 in this then u will always get a bigger number and it keeps increasing since it is being multiplied by a number always greater than one hence the series is divergent u can even try this by multiplying 10 by 1.1 u will realise the answer is always greater than 10 and it will keep increasing but lets look at a case where -1 <n<1 then u will realise that the coefficient will tend to reduce and they will converge towards a number that means the expansion will never stop it will keep increasing coz the one of those brackets in the coefficient equation will never be zero so hence there will be a number and this will happen for all non integer values but a series will converge (keep moving to a certain number but never reach that number) when |(whatever is in the bracket)| < 0 and then to find find the values of x for which the series is convergent you just make sure x remains on the left hand side and u end up with a value on the right hand side for example in the expansion (1 + x)^0.5 the series is convergent when |1 + x| < 0 then we make x the subject by taking 1 the other side but since it's in modulus it won't change the sign hence |x| < 1 which means the series is convergent for all values of x between -1 and 1.
    hope this helps if u don't understand a part let me know
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    (Original post by brainmaster)
    the reason we say that a the series is not convergent (it is divergent) when n>1 is because lets see the co efficient of x^3 in the expansion of (x+)^5;
    (n (n-1)(n-1))/3! ...if u try to replace any values of n>1 in this then u will always get a bigger number and it keeps increasing since it is being multiplied by a number always greater than one hence the series is divergent u can even try this by multiplying 10 by 1.1 u will realise the answer is always greater than 10 and it will keep increasing but lets look at a case where -1 <n<1 then u will realise that the coefficient will tend to reduce and they will converge towards a number that means the expansion will never stop it will keep increasing coz the one of those brackets in the coefficient equation will never be zero so hence there will be a number and this will happen for all non integer values but a series will converge (keep moving to a certain number but never reach that number) when |(whatever is in the bracket)| < 0 and then to find find the values of x for which the series is convergent you just make sure x remains on the left hand side and u end up with a value on the right hand side for example in the expansion (1 + x)^0.5 the series is convergent when |1 + x| < 0 then we make x the subject by taking 1 the other side but since it's in modulus it won't change the sign hence |x| < 1 which means the series is convergent for all values of x between -1 and 1.
    hope this helps if u don't understand a part let me know
    Ok honestly i no longer know if i even understand what validity and convergence are in binomial expansions. Doesnt convergence describe the situation when a value derived from substituting an X value into an infinite expansion , approaches but never becomes the value we get when we substitute X into the non-expanded form of the equation ? And what does validity describe in binomial expansions ?
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    Is validity the range of X values for which an infinite binomial expansion is convergent ? And outside that range the infinite expansion doesnt converge towards an approximate value and is invalid ?
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    (Original post by brainmaster)
    the reason we say that a the series is not convergent (it is divergent) when n>1 is because lets see the co efficient of x^3 in the expansion of (x+)^5;
    (n (n-1)(n-1))/3! ...if u try to replace any values of n>1 in this then u will always get a bigger number and it keeps increasing since it is being multiplied by a number always greater than one hence the series is divergent u can even try this by multiplying 10 by 1.1 u will realise the answer is always greater than 10 and it will keep increasing but lets look at a case where -1 <n<1 then u will realise that the coefficient will tend to reduce and they will converge towards a number that means the expansion will never stop it will keep increasing coz the one of those brackets in the coefficient equation will never be zero so hence there will be a number and this will happen for all non integer values but a series will converge (keep moving to a certain number but never reach that number) when |(whatever is in the bracket)| < 0 and then to find find the values of x for which the series is convergent you just make sure x remains on the left hand side and u end up with a value on the right hand side for example in the expansion (1 + x)^0.5 the series is convergent when |1 + x| < 0 then we make x the subject by taking 1 the other side but since it's in modulus it won't change the sign hence |x| < 1 which means the series is convergent for all values of x between -1 and 1.
    hope this helps if u don't understand a part let me know
    Btw please try to answer my questions with a yes or no where possible. It just makes understanding things easier and forget about the two questions i posted in the very first post on this thread. Thanks !
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    (Original post by Sam19KY)
    Ok honestly i no longer know if i even understand what validity and convergence are in binomial expansions. Doesnt convergence describe the situation when a value derived from substituting an X value into an infinite expansion , approaches but never becomes the value we get when we substitute X into the non-expanded form of the equation ? And what does validity describe in binomial expansions ?
    a convergent series means that when u expand it then then it converges (it gets close to a value but never reaches it) and this only happens for a certain values of x and then we say an expansion is infinite because it will continue forever and ever since n in not a positive integer.now when u ask validity, u will be asked questions like "for what values of x is the expansion valid?" now u see at the start they tell u that the expansion is convergent and as I said the expansion is only convergent for certain values of x so when they ask what values of x makes the expansion valid they r simply asking for what values of x is the series convergent.
    convergent series r used for approximation like if u want to approximate sqrt {5} then u can use convergent series.
    another thing u might be asked will be why can one value of x in the same series not give an approximation and the other value of x will give an approximation? now I just told u that approximation can only be made when the expansion is convergent so you will first find the range of x values that make the expansion convergent then u will realise that one value falls in the this range and the other value of x doesn't fall in this range so this tells that if we use that value of x Then we won't get an approximation since the series won't be convergent...
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    (Original post by brainmaster)
    a convergent series means that when u expand it then then it converges (it gets close to a value but never reaches it) and this only happens for a certain values of x and then we say an expansion is infinite because it will continue forever and ever since n in not a positive integer.now when u ask validity, u will be asked questions like "for what values of x is the expansion valid?" now u see at the start they tell u that the expansion is convergent and as I said the expansion is only convergent for certain values of x so when they ask what values of x makes the expansion valid they r simply asking for what values of x is the series convergent.
    convergent series r used for approximation like if u want to approximate sqrt {5} then u can use convergent series.
    another thing u might be asked will be why can one value of x in the same series not give an approximation and the other value of x will give an approximation? now I just told u that approximation can only be made when the expansion is convergent so you will first find the range of x values that make the expansion convergent then u will realise that one value falls in the this range and the other value of x doesn't fall in this range so this tells that if we use that value of x Then we won't get an approximation since the series won't be convergent...
    Hell yeah thanks man i got it !
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    (Original post by Sam19KY)
    Hell yeah thanks man i got it !
    that's great!! I struggled with understanding this too but I practised and tried making sense by what they mean valid and convergent however I havent done C4 or any core maths. I got this knowledge from my pure maths. I'm going to start A Levels in September
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    (Original post by Sam19KY)
    Btw please try to answer my questions with a yes or no where possible. It just makes understanding things easier and forget about the two questions i posted in the very first post on this thread. Thanks !
    I would but I don't want u to know that only if u understand the whole thing well then who knows u could come with different ways of solving binomial expansions that's y i prefer understanding a thing completely since it sticks to our brain
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