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# Number Theory/Taylor and binomial expansions

Question
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,
ML

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.

Regards

vijilant
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Number Theory

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#### Vijilant

##### Expertise

Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.

##### Experience

Teacher of math for 53 years

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AQA Doncaster Bridge Club Danum Strings Orchestra Doncaster Conservative Club Danum Strings Orchestra Simply Voices Choir Doncaster TNS mystery shopping St Paul's Music Group Cantley

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Journal of mathematics and its applications M500 magazine

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BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

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State Scholarship 1955 Highest Score in Yorkshire on OU course MST209 £50 prize First class honours in OU BA Mathematics

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I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things. One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.