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What is a function with domain $(−∞, ∞)$ but that does not have a Taylor series centered at $0$? Why doesn’t this function have a Taylor series centered at $0$?

I tried $\ln(x)$ but the domain was wrong. Someone please help, I really don't understand Taylor series very well.

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  • $\begingroup$ Any function that is not infinitely differentiable at the origin will not have a Taylor series about $0$. $\endgroup$
    – saulspatz
    Commented Apr 24, 2020 at 1:37

2 Answers 2

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The canonical example is $f(x):=e^\frac{-1}{x^2}$, with $f(0)=0$. You can prove that $f^{(n)}(0)=0$ (the $n$th derivative) for every $n$ by induction. Thus, if $f$ had a Taylor series about $0$, we would have $$ f(x)=\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n=0 $$ for every $x$ in some neighbourhood of $0$, which is absurd since $f(x)=0$ only for $x=0$.

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  • $\begingroup$ It has a Taylor series, but the series doesn't converge to the function. $\endgroup$
    – saulspatz
    Commented Apr 24, 2020 at 1:36
  • $\begingroup$ @saulspatz Oh you're right, I suppose I misread what the question was asking. I'll leave it up in case the OP finds it useful. $\endgroup$
    – Reveillark
    Commented Apr 24, 2020 at 1:37
  • $\begingroup$ Yes, I read the question that way too, at first, because taken literally, it seems awfully simple. $\endgroup$
    – saulspatz
    Commented Apr 24, 2020 at 1:38
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$f(x)=|x|$

Domain is all real numbers, but derivative does not exist at 0.

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