Expansion of $e^x$ - correct form I have come across in a textbook to an expansion of e to the x in the following form:
$$
1+ \frac1x + \frac1{x^2} + \frac1{x^3} + \ldots
$$
Is the above correct or is it a typo?
I am familiar with this type of expansion for e to the x:
$$
1+ \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots
$$
Are they both correct? If yes, how is the first one above arrived at? I could not find it online anywhere.
Thank you in advance.

Note the first series has a ratio of terms of $1/x$, thus, assuming $|1/x| < 1$,
$$
\sum_{k=0}^\infty \frac1{x^k}
 = \sum_{k=0}^\infty (1/x)^k
 = \frac{1}{1-(1/x)}
 = \frac{x}{x-1}
 = 1 + \frac{1}{x-1} \ne e^x.
$$
