Question about an infinite sum in which every term is infinite Given the sequence $x_k=\lim_{n \rightarrow 0} \frac{1}{kn^2}$ where $k\in\Bbb{N}$, define an infinite sum $S=x_1-x_1+x_2-x_2+x_3-x_3+...$ Now every $x_k$ is infinite, but does $S=0$? I know we cannot say $S=0$ but I don't understand why. $S$ can be rearranged as $S=(x_1-x_1)+(x_2-x_2)+(x_3-x_3)+...$ If we just look at the symbols, does not $x_1-x_1=0$, $x_2-x_2=0$, ... always because the minuend and subtrahend are identical? Why they cannot hold when $x_k$ is infinite?

Your sum $S$ is not well defined. As you say none of the terms in it are well defined. You can get the same behavior from well defined series that are convergent, but not absolutely convergent. A series is absolutely convergent if the series formed by taking the absolute value of all the terms is still convergent. So if we have the series $\log 2 = 1-\frac 12 + \frac 13 -\frac 14 \dots$. We know the harmonic series $1+\frac 12 + \frac 13 +\frac 14 \dots$ diverges so if we sum the terms from $\log 2$ in a different order we can get any answer we want. Yes, this says that for non-absolutely convergent series your intuition is not correct. Often our experience with finite objects (like sums) is not correct when we make the transition to infinite objects (like series).