Bounding error for non-alternating Taylor expansion

Question

Given a function whose Taylor Series Expansion is ${\textit not}$ an alternating series, such as $f(x)=e^x$, my understanding is that bounding the error term $E_{n+1}$ is not good enough to ensure accuracy. How do I bound the error so that I can approximate $f(x)=e^x$ say, up to 5 decimal places? Thanks.