How would you expand this exactly? Would you use the binomial theorem?
So begin by finding a few derivatives of the function, and evaluate them at zero. Then plug them into the form you see.
The binomial expansions says that
If n is a positive integer then
is defined as and is 0 for i> n so the sum terminates.
If n is not a positive integer, then n! is not defined but we can still right out the last of those- but the sum no longer terminates.
In particular, .
Perhaps the OP is referring to the representation of a binomial as an infinite sum: Binomial Theorem
for any and if . Here, let and .
So we have:
See where you can go from here. Try to simplify the general term and hopefully you'll be able to get somewhere