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C4 Binomial expansion - values for which expansions are valid? watch

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    My textbook goes on about finding the values for which the expansion is valid, usually mod x < 1, but I have no idea how to get it or what it means... Could anyone explain what it means and how do I find it out for a particular expansion?

    For example, (1-x)^1/3

    How would you know what values it is valid for?
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    (Original post by lollage123)
    My textbook goes on about finding the values for which the expansion is valid, usually mod x < 1, but I have no idea how to get it or what it means... Could anyone explain what it means and how do I find it out for a particular expansion?

    For example, (1-x)^1/3

    How would you know what values it is valid for?
    Take: \ (1+ax)^n Here the ax is the "x" for which |x| &lt; 1
    So:
    |ax| &lt; 1 \Rightarrow x&lt;...?
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    (Original post by lollage123)
    My textbook goes on about finding the values for which the expansion is valid, usually mod x < 1, but I have no idea how to get it or what it means... Could anyone explain what it means and how do I find it out for a particular expansion?

    For example, (1-x)^1/3

    How would you know what values it is valid for?
    The reason is that the series for (1+x)^n goes like:
    1 + nx+\frac{n(n-1)}2x^2+…
    If mod x >= 1, it's clear that this will never converge because x^2 > x, x^3 > x^2, …
    But if mod x < 1, it has a chance of converging because x^2 < x, etc - the terms are getting smaller. It can be shown rigorously that indeed the series converges for mod x < 1 (but that's a big kettle of fish involving analysis and the ratio test).
 
 
 
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