When approximating $e^{.1}$ using McLauren expansion the error has $e^{.1}$ - does this mean there is an error on the error?

Question

When approximating $e^{.1}$ using McLauren expansion, with $n=4$, the error is $\frac{e^{.1}(.1)^{5}}{5!}$. The expression $e^.1$ is part of the error. Does this mean there is an error to the error? And then another error for that... in a loop that would just keep going?

Answer 1

Actually the error is less than the expression. To get a usable bound, you need to replace $e^{0.1}$ by something you know to be larger. Say by $2$. Then you'll know your error is less than

$$\frac{2(0.2)^5}{5!} = \frac{1}{6000000}.$$