Why is the commutative property not true for conditionally convergent series?

[member 508213]

I know when a series is conditionally convergent and I understand that being conditionally convergent means that rearrangement of the terms will not always lead to the same sum, but I am unsure why exactly this is important? I have not really experienced a time where I am rearranging terms anyway (I am in Calc BC so maybe later on I will but not now). I am studying Taylor Series right now and it seems that conditionally convergent series still converge even if it is not "absolutely". I'm worried I'm missing some large concept because I do not see a big importance if something is conditional or absolute. Thanks for any explanations, I have tried to look online but everything I find seems to only explain how to determine conditional convergence and not the importance (other than explaining the rearrangement of terms part).

 

[Mark44]

I know when a series is conditionally convergent and I understand that being conditionally convergent means that rearrangement of the terms will not always lead to the same sum, but I am unsure why exactly this is important? I have not really experienced a time where I am rearranging terms anyway (I am in Calc BC so maybe later on I will but not now). I am studying Taylor Series right now and it seems that conditionally convergent series still converge even if it is not "absolutely". I'm worried I'm missing some large concept because I do not see a big importance if something is conditional or absolute. Thanks for any explanations, I have tried to look online but everything I find seems to only explain how to determine conditional convergence and not the importance (other than explaining the rearrangement of terms part).

The series 1 - 1/2 + 1/3 - 1/4 +- ... +(-1)^{n + 1}(1/n) + ... converges. The convergence is conditional because the series made up of the absolute values of the terms of this series diverges (1 + 1/2 + 1/3 + ... + 1/n + ... is the well-known harmonic series). Absolute convergence is a stronger form of convergence, as it implies the convergence not only of the series in question, but of another series, the one made up of the abs. values of the series in question.

 

[pwsnafu]

I know when a series is conditionally convergent and I understand that being conditionally convergent means that rearrangement of the terms will not always lead to the same sum, but I am unsure why exactly this is important?

The most important aspect is that it is counter intuitive. Finite addition is commutative $x+y=y+x$. Conditionally convergent series are not, so this is an example of a property which is true finitely but not infinitely.
Note that "will not always lead to the same sum" isn't the impressive aspect. Any conditionally convergent series can be rearranged to evaluate to any real number. Think about that, I can give you a conditionally convergent series and tell you "rearrange to sum to $x$" and you will always succeed. It means that the sum of a conditionally convergent series is completely determined by the order.