Taylor Series (Calculus II)

Date: 2/14/96 at 18:57:10
From: Anonymous
Subject: Taylor Series (Calculus II)

I don't have a real specific question. I am having difficulty 
understanding the Taylor Series and Maclauren Series. I guess I am 
having trouble with all areas of the Thomas, Finney 8th edition 
Calculus textbook's eighth chapter. 

I was looking for some help on the theory behind the error estimate 
and "c" in particular - I am not understanding it. I am looking for a 
really super guide that would help me better understand Calc II and 
where I might get this. My professor just does examples and doesn't 
explain the theory. 

Thank you very much. I'm sure you won't know how much I appreciate 
this.

Date: 7/29/96 at 13:41:47
From: Doctor Jerry
Subject: Re: Taylor Series (Calculus II)

The c in the Mean-value Theorem

f(b)-f(a)=f'(c)(b-a)

is a point between a and b, which (for smooth functions) is known to 
exist but is not necessarily easy to calculate.  Geometrically, it's 
quite clear that there must be a value of x, call it c, between a and 
b for which the slope of the line joining (a,f(a)) and (b,f(b)) is 
exactly equal to the slope of the graph of f at (x,f(x)).  

The c in Maclaurin's formula   f(a+h) =
f(a)+f'(a)x/1!+f''(a)x^2/2!+...+fn'(a)x^n/n!+fn+1'(c)x^(n+1)/(n+1)!
where fn' means the nth derivative and fn+1' means the (n+1)st 
derivative, is precisely analogous to the c in the Mean-value Theorem.  

The proof that it exists is more complicated, but not different in 
kind.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/