I have always been puzzled by the phenomenon:
We know that $\mathit{e} = \sum_{k=0}^\infty \frac{1}{k!}$, and let $s_n= \sum_{k=0}^n \frac{1}{k!}$ the partial sum of $\mathit{e}$. We also know that this series converges.
I saw in some proof you can actually take $\mathit{e} - s_n= \sum_{k=n+1}^\infty \frac{1}{k!}$. I always thought that the $\sum_{k=0}^\infty$ is just a symbol that signifies that you let the partial sum tends to infinity and that's it so I do not really know how to justify the operation of $\mathit{e} - s_n$.