# Taylor series error term - graphical representation

1. Nov 3, 2012

### wizkhal

Hello all,
Recently I've found something very interesting concerning Taylor series.
It's a graphical representation of a second order error bound of the series.
Here is the link: http://www.karlscalculus.org/l8_4-1.html [Broken]

My question is: is it possible to represent higher order error bounds in a similar way?
For example: third order error term would have "3! = 6" in a denominator...
I know that Taylor series is based on Mean Value Theorem and I know the proof of it.
However it would become much clearer if it was possible to represent error bounds in a graphical way.

Have a nice weekend.

Last edited by a moderator: May 6, 2017
2. Nov 3, 2012

### lurflurf

Yes, see any calculus book, or this

http://mathworld.wolfram.com/SchloemilchRemainder.html

It follows from the mean value theorem

Often in simple examples the error is well approximated by the next term as in

$$\sin(x) \sim \sum_{k=0}^\infty \frac{x^k}{k!} \sin\left( x+k \frac{\pi}{2}\right)$$

Last edited: Nov 3, 2012