exponential type series question

this looks really simple but might end up quite complicated. i'm currently stuck though :/

suppose that **for every k**, $\displaystyle \sum_{n=0}^\infty \frac{a_{k+n}}{n!}=e$. does this imply that $\displaystyle a_k \to 1$?

the above condition can also be rewritten as $\displaystyle \sum_{n=0}^\infty \frac{1-a_{k+n}}{n!}=0$ of course, which makes it even more painfully obvious, but I still can't prove this...

I need also to figure out a more general question: does $\displaystyle \lim_{k\to\infty}\sum_{n=0}^\infty \frac{a_{k+n}}{n!}=e$ imply that $\displaystyle a_k \to 1$, but clearly knowing the answer for the initial question would be a good start...

any partial ideas or references will be appreciated too!..