# Question on bounding error tems in Taylor's Theorem

I have a quick question about Taylor's Theorem.

Questions

For an infinitely differentiable function, if the error term in Taylor's Theorem tends to zero, does the existance of a Taylor series follow from the fact the series and the partial sum to N can differ by at most $$R_N$$ (where this is the error from Taylor's Theorem), which tends to 0, meaning we can make the difference as small as we like and hence they are the same?

I hope that is clear. If not please let me know. Feel free to also post short comments as answers.

• No, this is not clear. Error term tends to zero when $x$ tends to $x_0$ or when $N \to \infty$? Use mathjax to state your question clearly. May 27, 2022 at 22:22

My understanding of your question is: Assuming that $$f(x)$$ is infinitely differentiable, and $$f(x_0) - P_n(x_0) \to 0$$ as $$n \to \infty$$, where $$P_n$$ is the $$n^{th}$$ Taylor polynomial for $$f(x)$$ at $$x_0$$, can we conclude that $$f$$ has a Taylor expansion at $$x_0$$?
The page linked above gives a classic example of an infinitely differentiable function all of whose derivatives are $$0$$ at $$x = 0$$, but the function is not equal to $$0$$ in any neighborhood of $$x = 0$$. All of the Taylor polynomials will therefore be $$0$$ at $$x = 0$$, so the difference $$f(0) - P_n(0) = 0$$ for all $$n$$. On the other hand, $$f(x) \neq 0$$ on any neighborhood of $$0$$, so $$f(x)$$ can't be equal to its Taylor expansion at that point.