How is it that some forms of an equation can have a defined outcome while other forms of the same equation are undefined? $\newcommand{\kinv}{(\frac{1}{k})}$
Take the equation $S_k = \frac{k-1}{k!}(\kinv^0+\kinv^1+\kinv^2+\cdots)$. Evaluate it for $k=1$, and we can see that $k-1=1-1=0$, so the whole thing must equal $0$.
Good. Now solve the geometric series and we find that $S_k = \frac{k-1}{k!}(\frac{k}{k-1})$. But if we evaluate this at $k=1$, then $S_k$ is undefined because we have $0$ in the denominator!
Okay, so what to do? Let's simplify to this: $S_k = \frac{k}{k!}$. So now we can evaluate it once again, and it is not undefined, but rather $1/1!=1$. Wait, what? Our previous, equivalent formula gave us $0$, not $1$!
Okay, what if we cancel the k's? Then we'd have $S_k = \frac{k}{k(k-1)(k-2)\cdots} = \frac{1}{(k-1)!}$. And here, once again, we have a $0$ in the denominator if we try to evaluate, and so the solution is once again undefined. :-(
Can someone explain why these equations don't always evaluate the same, even though they are valid and equivalent?
Correction: As others have pointed out, the 4th iteration of the formula actually evaluates to $1$.
 A: They don't evaluate differently.  If you plug in $k=1$ into the first expression, the $k-1\over k!$ becomes 0, as you said, but that does not mean you can conclude that the product is 0, because the $\left(\left({1\over k}\right)^0 + \left({1\over k}\right)^1 + \left({1\over k}\right)^2 + \cdots \right)$ becomes $1+1+1\cdots = \infty$.  The whole expression is therefore equal to $0\cdot\infty$, which is, guess what, undefined, because of exactly this sort of situation.
This kind of situation comes up all over the place. For example, consider the infinite summation: $$\sum{1-1+1-1+\cdots}$$
Grouping the terms one way, they all cancel: $$(1-1)+(1-1)+\cdots = 0$$
But grouping them differently, the sum seems to be 1: $$1-(1-1)-(1-1)-\cdots=1$$
The conclusion is that the series can't be handled sensibly or consistently, and we say that this "summation" is indeterminate.  A large part of the branch of mathematics known as analysis is devoted to determining when such questions make sense and can be handled consistently.
