Use a loop to create each term, summing the result as you go. A while loop allows you to do this, with a test inside the while statement to see if the terms are sufficiently small. Or, use a for loop, with a test inside the loop, then a call to break when the terms are sufficiently small.
Compute the terms using the log of each terms, then exponentiate each term before you sum it into the running sum. Gammaln here will be efficient to compute the log of the gamma function, but also the log of the factorial. The avoids partial underflows and overflows inside the terms because you use logs.
Beyond that, just start writing. The way to learn is to write the code yourself. If I do it for you, then you learn nothing.
