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    Hey there, i just wanted to ask a question about this sentence I read:

    "If n is not a positive integer, i.e it is fractional or negative, then the series is infinite and the expansion will only be valid for |x|<1 "

    Where n is the power eg (1+x)^n

    I understand why the series will be infinite, but why will the expansion only be valid for |x|<1 ??

    Thanks if you can help
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    (Original post by Nice.Guy)
    Hey there, i just wanted to ask a question about this sentence I read:

    "If n is not a positive integer, i.e it is fractional or negative, then the series is infinite and the expansion will only be valid for |x|<1 "

    Where n is the power eg (1+x)^n

    I understand why the series will be infinite, but why will the expansion only be valid for |x|<1 ??

    Thanks if you can help
    Look at the individual terms. Imagine x = 2. The terms in the expansion would get larger and larger so the series would diverge.
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    (Original post by Mr M)
    Look at the individual terms. Imagine x = 2. The terms in the expansion would get larger and larger so the series would diverge.
    Ahh I didn't see the problem with that until now - the series MUST be finite, right? If the series is infinite, the binomial expansion will not work?
    Well, won't be valid at least
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    (Original post by Nice.Guy)
    Ahh I didn't see the problem with that until now - the series MUST be finite, right? If the series is infinite, the binomial expansion will not work?
    Well, won't be valid at least
    It will if |x| < 1 as the terms become increasingly small and less significant.
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    (Original post by Mr M)
    It will if |x| < 1 as the terms become increasingly small and less significant.
    Oh yeah, but in that case you can treat it like a finite series because it converges, yes?
    Thanks for your help it's clicked I just like to understand exactly what's going on and why, even if my questions come across as stupid/irrelevant
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    (Original post by Nice.Guy)
    Oh yeah, but in that case you can treat it like a finite series because it converges, yes?
    Thanks for your help it's clicked I just like to understand exactly what's going on and why, even if my questions come across as stupid/irrelevant
    You can't "treat it like a finite series" because it isn't - it's an infinite series!

    What you can say is that because the terms get smaller and smaller, if you stop after a certain point then you can get the answer to any desired accuracy.
 
 
 
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