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Let $f:(-\infty, 1) \to \mathbb{R}$ be a function, $f(x)=e^x+ln(1-x)$.

Find $n \in \mathbb{N}$ such that the error when approximating $e^{1.1}+\ln(1.1)$ by their Taylor polynomial $T_{n,f,0}(x)$, with $n < 0.0001$.

I'm really confused and I have no idea where to start.

I know that $f(x)=T_{n,f,0}(x)+R_{n}(x)$ where $R_{n}(x)$ is the reminder of the Taylor polynomial.

$e^{1.1}+\ln(1.1)=(1-\frac{x^3}{6}-\frac{5 x^4}{24}-\frac{23 x^5}{120}-...)-n$ Where $n$ is the given error?

Sorry if this is so confusing but I'm really lost!

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  • $\begingroup$ Should "$t \in \mathbb{N}$" be "$n \in \mathbb{N}$"? $\endgroup$ – angryavian Nov 23 '18 at 21:29
  • $\begingroup$ Yes! I got confused with another problem. I will fix it now. $\endgroup$ – Moria Nov 23 '18 at 21:30
  • $\begingroup$ Don't just blindly replace all $t$s with $n$s; now $n < 0.0001$ does not make any sense. Presumably you want to find $n$ such that the error of the Taylor polynomial $T_{n,f,0}$ is $< 0.0001$. $\endgroup$ – angryavian Nov 23 '18 at 21:32
  • $\begingroup$ It must be a typo, it was written like that in an old exam. But that's what I have to do and I literally have no idea where to start. $\endgroup$ – Moria Nov 23 '18 at 21:37
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Your $f(x)$ does not match the function you are asked to form the Taylor series of because it has $\ln (1-x)$ and the $e^x$ term is $0.1$, not $1.1$ when $x=0.1$. Your Taylor series is not correct-it should have an $x$ on the left, and the Taylor series of $e^{1+x}+\ln(1+x)$ is not what you have shown-at $x=0$ it should be $e$, not $1$.

Once you get the right Taylor series, $n$ is the number of terms you need to keep to get the error at $x=0.1$ to be less than $0.0001$. You can look up the remainder term of the Taylor series, which will be $0.1^{n+1}$ times the $n+1^{st}$ derivative of the function. You can bound the $n+1^{st}$ derivative in the interval $[0,0.1]$ and evaluate the upper bound for the error this gives you.

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  • $\begingroup$ Thank you!! I'll have to ask my teacher about it. It was written like that in an old exam. Now it's clearer how to solve this problem. $\endgroup$ – Moria Nov 23 '18 at 22:04

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