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?


Why does the order of summation of the terms of an infinite series influence its value ?

Because, although the series is convergent, it is not absolutely convergent.
