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Intuitively, why does the Maclaurin expansion work? Watch

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    (Original post by DFranklin)
    For an intuitive explanation, imagine we have two balls. If they start at the same place, with the same initial speed, and the same acceleration, and the same rate at which the acceleration changes, and the same rate at which the rate of change of the acceleration changes etc..., then it's intuitively believable that they will move in exactly the same way.

    Unfortunately, as the counterexample I posted shows, this turns out to not actually be the case. (In terms of the integral proof I showed you, the issue is that the final "integral" term doesn't get small enough to be ignored, no matter how many terms you take).

    So at this point in your mathematical career, you're faced with something that you'd like intuitive justification for, and I can even given you a plausible intuitive argument, and yet it isn't actually true.

    And then at university, when you study integration with complex numbers (not sure if you'll have done complex numbers at all yet), you find that if you have a complex function, and it's (complex) differentiable just once (for every point in a region), then you can make a Taylor expansion about any point in the region and all the derivatives will work and it will come out perfectly.

    And of course at this point, your scepticism will have been tuned to the point where you really wouldn't expect this to work, and so your intuition gets fooled all over again...

    Gotta love maths!
    Indeed! And I've just looked in my textbook and found a chapter near the end of FP3 entitled 'Calculus with complex numbers'. So it looks like some of the fun will begin before I even get to uni :-)
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    (Original post by peterw55)
    Indeed! And I've just looked in my textbook and found a chapter near the end of FP3 entitled 'Calculus with complex numbers'. So it looks like some of the fun will begin before I even get to uni :-)
    You won't do the interesting stuff until university I'm afraid (complex integration is usually a 2nd or 3rd year topic, in fact)!
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    (Original post by DFranklin)
    You won't do the interesting stuff until university I'm afraid (complex integration is usually a 2nd or 3rd year topic, in fact)!
    Really enjoyed complex analysis, had to get contour integration up to scratch first and then proved Taylor's Theorem and the Laurent Series using Cauchy's Integral Theorem.
 
 
 
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