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Number Theory/Taylor and binomial expansions

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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

Answer
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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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

Organizations
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

Publications
Journal of mathematics and its applications M500 magazine

Education/Credentials
BSc (Hons) Liverpool (Science). BA (Hons) OU (Mathematics)

Awards and Honors
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.

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