What is the integral of $\frac{1}{k!}$?

Question

I have been struggling with the following problem in probability:

Assume X is a random variable with the following probability density function:

$$ P(X = k) = \frac{A}{k!}, k=0,1,2,... $$

How to find the coefficient A ?

My first thought was to integrate this function to get its cumulative distribution function, and then make it equal to 1. However, I find out that integrating such a function is not so simple.

Is it possible to solve it in a simple way?

Thanks!

Answer 1

This is a discrete distribution, so instead of integrating, try summing over $k$:

$$\sum_{k=0}^\infty P(X=k) = 1$$