Why does the order of summation of the terms of an infinite series influence its value? I was looking through my lecture notes and got puzzled by the following fact: if we want to find the value of some infinite series we are allowed to rearrange only the finite number of its terms. To visualize this consider the alternating harmonic series:
$$\sum_{n=1}^\infty(-1)^{k-1}\frac1k=1-\frac12+\frac13-\frac14+\frac15-+\dots=0.693147...$$
But if we rearrange the terms as follows the value of the series gets influenced by this action:
$$1+\frac13-\frac12+\frac15+\frac17-\frac14+\dots=1.03972...$$
So commutativity of addition isn't true on infinity? How was it obtained and how can it be proved?

It's quite easy to think up elementary counter-examples.
For example, consider the series
$$1-1+1-1+1-1+...=(1-1)+(1-1)+(1-1)+...\\
=0+0+0+...\\
=0.$$
If it is permissible to commute an infinite number of terms, you can rearrange the series into,
$$1-1+1-1+1-1+1-...=1+(-1+1)+(-1+1)+(-1+1)\\
=1+0+0+0+...\\
=1,$$
implying $0=1$. Generally speaking, $0=1$ is undesirable result.
