Remainder Theorem of Taylor Polynomials

[Ohoneo]

My professor didn't really explain this well, and I can't find much additional information online, and of course, the textbook uses only the easiest of examples.

Find the remainder term in the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = e^-x; a = 0

I got as far as deriving it all the way to the fourth derivative, but I honestly have no idea where to go from there. Any help is appreciated!

 

[galactus]

Lagrange's form of the remainder:

\(\displaystyle \displaystyle f^{(n+1)}(x)=(-1)^{n+1}e^{-x}, \;\ R_{n}(x)=\frac{(-1)^{n+1}e^{-c}}{(n+1)!}x^{n+1}\)

 

[Ohoneo]

Thank you! So my other question is that part of the answer states "for some c between x and 0," why is it between x and 0?

 

[Deleted member 4993]

Thank you! So my other question is that part of the answer states "for some c between x and 0," why is it between x and 0?

You need to review the process of derivation of Taylor's Series Expansion.

What is the radius of expansion?

The answer to your question will be self-evident when you investigate the derivation.

 

[Ohoneo]

I actually figured it out as soon as I posted this, funnily enough. Thank you so much for all of your help!