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?

Solution 1:

We have the strong intuition that changing the order of the terms in a sum never changes the sum because it never does for a finite sum. We didn't just accept without proof that it never does for a finite sum. Rather, it's a theorem that can be proven from the associativity and commutativity of addition despite the fact that the associative law only states associativity for a sum of 3 terms and the commutative law only states commutativity for a sum of 2 terms. That proof can't be generalized to the case of an infinite sum. Note that the positive terms in the first series add to infinity and the negative terms in it add to negative infinity. The reason the 2 series have different sums is because as n gets larger, the sum of the negative terms within the first n terms of the second series never gets anywhere near keeping up with the sum of the negative terms within the first n terms of the first series.

Solution 2:

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.