Math for Non-Geeks/Absolute convergence of a series

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In this article, you may learn about the absolute convergence, which is a stronger property than just convergence. Absolute convergence is able to guarantee that a series keeps the same limit under re-arrangement of elements. So an absolute convergent series has a kind of unique limit, independent of the order of summation. The concept also turns out useful for defining integrals ("=continuous sums") with unique values.

Motivation

Does the value of a sum depend on the order of the order of elements? For finite sums, the answer is no, as you may have already learned in school. For instance,

is the same as

This is a consequence of "commutativity of the sum": there is $a+b=b+a$ for all $a,b\in ℝ$ . Finitely many commutations of neighbouring elements will not change the value of the sum. However, for infinitely many commutations, it is not clear whether the value of the sum remains invariant. So may the value of the sum depend on the order, here? The answer turns out to be yes! A series like

might have a different value than its re-arranged version

It might even happen that a convergent sum gets divergent under re-arrangement and vice versa. Therefore, it might be useful to have a criterion for when the value of a series does not depend on the order of its elements. And indeed, such a criterion exists:

The idea is there are sequences which include both positive elements which add up to $\mathrm{\infty }$ and negative elements, which add up to $-\mathrm{\infty }$. The term $\mathrm{\infty }-\mathrm{\infty }$ is not defined and the limiting behaviours of such a series may depend on how elements are combined. For instance, consider the series

The partial sums of this series jump between 1 and 0, so they do not converge, but stay bounded. However, one may re-arrange the elements such that alternatingly two positive and two negative elements follow each other:

Since $1+1-1=1$, this series is equivalent to $1+1+1+\dots =\mathrm{\infty }$. Equivalently, by taking "more negative than positive elements", we can reach

How to avoid these ambiguities? It may be a good ansatz to require that the sums of absolutes of all positive and of all negative elements are smaller than infinity. Then, we do not run into the problem of having an ill-defined limit $\mathrm{\infty }-\mathrm{\infty }$. Having bounded sums of the absolute values is actually what characterizes an absolutely convergent series.

Definition

So a series is absolutely convergent, if and only if the series of the absolute values of its elements converges. This forbids the case, where all $a_k>0$ or all $a_k<0$ sum up to $\mathrm{\infty }$ or $-\mathrm{\infty }$.

Absolutely convergent series converge Template:Anchor

Absolute convergence is the strongest form of convergence. This sentence can be formulated as a mathematical theorem:

Not all convergent series converge absolutely

Every absolutely convergent series converges. However, the converse does not hold true (otherwise one would not need the notion of an absolutely convergent series). An example for a series which converges, but does not converge absolutely is the alternating harmonic series $\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{1}{k}$. Using the Alternating series test, one can prove its convergence. However, the series of absolute values $\sum _{k=1}|(-1{)}^k\frac{1}{k}|=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}$ is just the harmonic series, which is known to diverge. We conclude:

Criteria for absolute convergence Template:Anchor

Not every convergent series converge absolutely. Are there conditions under which a convergent series is known to converge absolutely? Indeed, there are! In order to find such a condition, we may ask the question how can a not absolutely convergent series possible converge? we can decompose such a series $\sum _{k=1}^{\mathrm{\infty }}{a}_k$ into its positive and negative parts:

and

for instance, for$a_k=(-1{)}^{k+1}\frac{1}{k^2}$, there is

and

We may think of ${\left(\sum _{k=1}^n{a}_k^{+}\right)}_{n\in \mathbb{N}}$ as the "budget of positive values" and of ${\left(\sum _{k=1}^n{a}_k^{-}\right)}_{n\in \mathbb{N}}$ as the "budget of negative values" of the series $\sum _{k=1}^{\mathrm{\infty }}{a}_k=\sum _{k=1}^{\mathrm{\infty }}(a_k^{+}+a_k^{-})$. Absolute convergence means that the combined budget $\sum _{k=1}^{\mathrm{\infty }}(a_k^{+}-a_k^{-})$ is finite. For absolutely divergent series, the budget must be infinite. Now, a convergent but not absolutely convergent series combines an infinite budget to a finite limit, by cancelling out infinities within the budget like $\mathrm{\infty }-\mathrm{\infty }$. We can prevent this by requiring that the "budgets" ${\left(\sum _{k=1}^n{a}_k^{+}\right)}_{n\in \mathbb{N}}$ and ${\left(\sum _{k=1}^n{a}_k^{-}\right)}_{n\in \mathbb{N}}$ of the series are finite:

Re-arrangement of series

In some cases, it might be convenient to re-arrange elements within a series. The important question is now: does the limit stay invariant under any re-arrangement? A mathematician divides series into two sets, depending on what the answer to that question is:

unconditional convergence

A series convergences unconditionally, if it has the same limit as any series obtained by re-arrangement of the elements.

conditional convergence

A series converges conditionally, if there are re-arrangements, under which have a different limit than the original series.

For real-valued sequences, a series converges unconditionally, if and only if it converges absolutely. The proof will be given in the next article Rearrangement theorem for series. The above notion of conditional and unconditional convergence becomes important for non-real valued series. then, absolute convergence might be not equivalent to unconditional convergence and one needs two different notions to distinguish both situations.

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