Cosmologicon wrote:For instance, how would you write the power series for exp(x) in sigma notation? Most people would answer Sum x^n/n!, n = 0...infinity, rather than 1 + Sum x^n/n!, n = 1...infinity.
Hmm, that's a good point. Although I would probably just write 1+x+x^2/2+x^3/3!+.... I've never really thought about that before. I guess I naturally just remove removable discontinuities. For example, if f(x)=e^(-1/x^2), then for me f(0)=0, although strictly speaking it should be undefined unless you've defined it using a piecewise construction. But it seems to me that when you're defining 0^0=1 in that sum you're not doing it because an indeterminant form is 1, you're doing it because this particular instance of the indeterminant form arises from taking the limit of x^0 as x approaches 0, which is obviously 1. You're not saying "we define the indeterminant form 0^0 to be 1", which would be silly, since it's an indeterminant form.
I think I was confused because it seemed you were saying that "in most cases we define all instances of this indeterminant form to be 1", but what you were really saying was "in most cases we define this indeterminant form to take on its limit, which is usually 1 when it arises", which is true since if you look at the limit of f(x)^g(x) as x tends to x_0, and f(x_0)=g(x_0)=0, you will get 1 if f and g are analytic, and f is not identically 0, which is usually the case in practice.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson