So, while trying to find something else, it looks like I've found, for many f(x)f(x):
f(x)+f′(x)+f″(x)+f(3)(x)+⋯+f(n)(x)
Assuming that there is an easy way to find this sum above, is there any use for it? I will elaborate a bit. I mean that I believe I have found a method that finds the sum of all of the derivatives above, and is much faster than calculating each derivative. In fact, it seems that calculating the sum above for most functions isn't much harder than calculating f(n)(x)f(n)(x), and it also should give a "closed form" of elementary expressions for most f(x)f(x).
I have one example that comes to mind: a "closed form" for a partial sum of exex, as in this question. If my ideas work, we would have the closed form that this question asks for.
So I'm wondering, is there anything else that this method is useful for?
IMPORTANT NOTE
I'm assuming that we have use of the "fractional calculus", which gives us the ability to calculate f(n)f(n) reasonably well and efficiently, using "differintegrals". This may make the sum above fairly trivial. I'm sorry if I misled anyone.
