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Number Theory/Taylor and binomial expansions


Hi Prof. Vj,
Just a curiosity: Is there a connection, in the sense of geometric interpretations, between the above two?
For example, a smooth real function y(x)^(1/2) can be expanded via both ways and the result are the same (at least to the first or 2nd order approximation, I think).
(Taylor's is easier to visualize, for each term what it implies.  But how can one relate them to a binomial expansion)
Hope this is not an absurd question : )
Kind regards,

Hello ML
The Taylor expansion needs the second term inside the bracket small.  So with the binomial, (c+d)^n, we can arrange for c to be larger than d.  Then take out the factor c^n to give
c^n(1 + d/c)^n. We can then write x = d/c.  And f(x) = x^n.  Then in Taylor's theorem, we choose a = 1. Then each derivative of f(x) evaluated at x=1 is a factorial and the result of the binomial appears.
Hope this is what you wanted.
Calculus is not really my subject, but I am willing to answer any math question.



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