Construct an expression that bounds the truncation error for a given value of x.

Question

The function that has been approximated with a Taylor Series is:

$$f(x) = \frac{log(x) - x + 1}{(x-1)^2} $$

It resulted in: $$f(x) = \sum_{n=2}^{\infty} \frac{(-1)^{k-1}}{k} (x-1)^{k-2}$$

I know truncation error is:

$E = f(x) - T_n (x)$

From this, I want to construct an expression that bounds the truncation error, assuming n > 2, for a given value of x.

How does one do this with a Taylor series approximation? Do you use Taylor remainders?

There is simply not enough information here to determine what your question is about. You need to add more details. — Carl Christian, Jan 24 '20 at 22:37
I have added more information, thank you for letting me know! — datanodes, Jan 25 '20 at 01:04

score 0 · Answer 1 · answered Jan 25 '20 at 08:14

0

You could use a geometric series as majorant, $$ |f(x)-T_m(x)|\le\frac1{m+1}\sum_{n=m+1}^\infty |x-1|^{n-2}=\frac1{m+1}\frac{|x-1|^{m+1}}{1-|x-1|}. $$

answered Jan 25 '20 at 08:14

Lutz Lehmann

126,666

Construct an expression that bounds the truncation error for a given value of x.

1 Answers1