Question : What is the distinction between the Taylor Series and the Taylor Polynomial? What are the restrictions on Taylor / MacLaurin Series'? What criteria must be met for a MacLaurin Series to be solved for if a Taylor Series is possible?
How would you explain to someone the train of thought and the progress in mathematics that took one mathematician from Taylor Series into and beyond the MacLaurin Series? How did all this happen?
I understand the math... I just dont understand the chronology ..
Answer : I think I understand your confusion. Perhaps a fellow mathematician can answer this better or correct me. Because I really dont know... I have a vague understanding, myself, as I have just been introduced to these subjects. I barely know calculus.
Anyway... you say you understand the proof for Taylor expansions... okay.
Realize, now, that ALL expansions form an infinite series. Its just that for most polynomials with 'normal' terms, the exponent reduces. Eventually, given enough differentiation, a function becomes a constant... and then a zero. This forces the terms beyond a certain point and into infinity to become zero. That cancels them out completely and reduces the size of the polynomial to a given number of terms.
I dont know what makes some infinite and some not... but any regular polynomial is its own Taylor expansion. And any function whose derivative is unchanging (like e^x) or whose derivative is cyclic (like sin x) makes no progress at reduction and therefo..
Question : I can't find anything!!!!!!!!!!!!!!!!!
Answer : I searched in Yahoo using the quotes as shown for:
'math history' cubic equations
In the 16th century in Italy, there occurred the first progress on polynomial equations beyond the quadratic case. The person credited with the solution of a cubic equation is Scipione del Ferro (1465-1526), who lectured in arithmetic and geometry at the University of Bologna from 1496 until 1526.
Question : Why is it important to simplify radical expressions before adding or subtracting? How is adding radical expressions similar to adding polynomial expressions? How is it different? What is an example of a radical expression?
Answer : Radical expressions act like indeterminants. If you have a simple expression like:
1/ ( (3) + (5) )
and you want to add it to:
1 / ( (3) + 7 )
it is very hard. You have to find a messy common denominator and do all sorts of junk. What a pain. But if you have:
(3/8) 5 + (1/4) (3)
and you wanted to add this to:
(1/4) 5 + (1/8) (3)
That's easy. Just add up the coefficients:
3/8 + 1/4 = 5/8
1/4 + 1/8 = 3/8
(5/8) 5 + (3/8) (3)
SO this is similar to adding polynomials because the radicals act like placeholders. In fancy math talk, extending rational numbers to include radical expressions forms a vector space with basis of the radical expressions. If you don't know what that means, ignore it.
So like polynomials, you have placeholders. Unlike polynomials, if you are working with (2) and (5), you may also encounter (10), but nothing else. You cannot obtain (3), and (2) becomes rational (for a polynomial, x is just a..