A) Find a polynomial approximation for $f(x)=2e^x$ centered at $0$ for values of $x$ in the interval $[-1,1]$

B) what is the actual bound on the error in your approximation given by Taylor theorem?

Hello, I have no clue on how to even start problems like this, assistance is appreciated.

  • $\begingroup$ What does Taylor's Theorem say first of all? $\endgroup$ – user417289 Nov 8 '17 at 20:29

The taylor series formula around $0$ is: $$f(x)=\sum_{n=0}^\infty f^n(0)\frac{x^n}{n!}$$ For $e^x$, computing each derivative is $1$ so:$$2e^x=2\sum_{n=0}^\infty \frac{x^n}{n!}$$.

Using Remainder Estimation Theorem: $$|R_{n}(x)|\le \frac{M|x|^{n+1}}{(n+1)!}$$ Where $M$ is the maximum value of $f^{n+1}(x)$ on the interval. We know that that is just $2e$, so $$|R_{n}(x)|\le \frac{2e|x|^{n+1}}{(n+1)!}$$


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