Give an example of a function that converges to its Taylor series for $x\geq 0$ but not for $x<0$.

Question

In Bartle Sherbert's analysis book,I found a problem.I have to construct a function such that it converges to its Taylor series on $[0,\infty)$ but not on $(-\infty,0)$.I am confused about how to approach this problem.Can someone suggest some idea.

Answer 1

Take $f(x) = e^{1/x}$ for $x<0$ and $f(x)=0$ for $x\ge 0.$ You can show that $f^{(n)}(0)=0$ for every $n\ge 0.$ (For this, you may proceed inductively and use L'hopital's rule.)

Hence, the Taylor series (centred at $0$) becomes just the zero function. So this Taylor series matches with $f(x)$ for every $x\ge 0$ but does not match for every $x<0.$