Question: The following series converges to exponential.

$\sum\_{i=0}^{\infty} \frac{x^i}{i!} = e^{x}$ 

Is the convergence of the following series known?

$\sum\_{i=0}^{\infty} \frac{(x-i)^i}{i!}$

Answer: 
This series diverges. In fact the terms $\frac{(x-i)^i}{i!}$ do not even tend to zero for any fixed $x$.
Your series is [asymptotically equivalent](https://artofproblemsolving.com/wiki/index.php?title=Asymptotic_equivalence) to 
$$
\sum\_{i=0}^\infty\frac{(-1)^ii^i}{i!}
$$
Absolute value of each term of the above series is greater than $1$ (except the first two terms) and hence it diverges and so does the original series.