Taylor Polynomials error bounds

**khuezy** · October 7th 2008, 07:56 PM

Hi, I got a question on the error bound.
lets say f(x) = (1)/(1-x) b=0 bound error (-1,1) and find where the error is <.001.

**ThePerfectHacker** · October 8th 2008, 09:46 AM

Originally Posted by khuezy

Hi, I got a question on the error bound.
lets say f(x) = (1)/(1-x) b=0 bound error (-1,1) and find where the error is <.001.

$1+x+x^2+...+x^n = \frac{1 - x^{n+1}}{1 - x} = \frac{1}{1-x} - \frac{x^{n+1}}{1 - x}$

Note that $1+x+x^2+...+x^n = T_n(x)$ - the $n$ th Taylor polynomial.

Therefore, $\left| T_n(x) - \frac{1}{1-x} \right| = \frac{|x|^{n+1}}{|1-x|}$

The error term $\frac{|x|^{n+1}}{|1-x|}$ cannot be bounded uniformly.
Only for a particular value of $x\in (-1,1)$ can you make the error sufficiently small.

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