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Question

I was wondering if one could define the sum of conditional convergence without using the notion of before or after (time)?

My Understanding

We define the following partial sum:

$$ S_n = a_1 + a_2 + a_3 + \dots + a_n $$

Then take the limit $n \to \infty$:

$$ \lim_{n \to \infty} S_n = \sum_n a_n$$ Now over here it seems to me this has a very specific meaning:

$$ S_n = ((((a_1 + a_2) + a_3) + \dots) $$

So we "first" add $a_1 + a_2$ "then" we add the resulting sum to next term $\underbrace{(a_1 + a_2)}_{S_1} + a_3$ and so on ... Of course one can map this series to a number but then the question arises why that particular choice of mapping.

I am slightly aware of other definitions of summations (such as Abel summation, etc) and was wondering if any of them had a formulation of the series without the notion of "time" (before/after).

Note: I know this is a slightly unusual question. I would be appreciate it if I could be guided on how to frame it better.

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    $\begingroup$ A related idea is the Riemann Rearrangement Theorem, which states that if you rearrange a conditionally convergent sum, you can make the limit anything that you choose. $\endgroup$ – Michael Burr May 21 '16 at 13:14
  • $\begingroup$ Are you asking if there's a way to define summation without the idea of sequence convergence? $\endgroup$ – AJY May 21 '16 at 13:14
  • $\begingroup$ @Y.Fan Can you then define the summation of series without using the words before/after/etc ... Or without stating that one must read from left to right? $\endgroup$ – drewdles May 21 '16 at 13:17
  • $\begingroup$ Sure. Let $s_{k + 1} = s_{k} + a_{k + 1}$. Then $\sum_{k = 1}^{\infty} a_k = \lim_{k \to \infty} s_k$. Words of order or sequence unused. $\endgroup$ – AJY May 21 '16 at 13:35
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    $\begingroup$ @AnantSaxena What? $\endgroup$ – Did May 22 '16 at 7:01
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If I am interpreting your question correctly, what you refer to as "a notion of time" is really just an order of summation. The key property of time that you are using is that it has a definite direction ("the arrow of time"), just like the order in which the partial sums are developed for a conditionally convergent series.

https://www.math.washington.edu/~morrow/335_14/history%20of%20rearrangements.pdf

As @MichaelBurr points out, for conditionally convergent series, this is impossible -- the terms are completely non-associative, i.e. the answer is entirely dependent on the order of the terms, since appropriate rearrangements of the terms can produce arbitrary limits for the series.

You don't need a different type of convergence to have the result be independent of the order (i.e. "not using the notion of time"); you just need to have an absolutely convergent series.

https://en.wikipedia.org/wiki/Absolute_convergence

Specifically, for an absolutely convergent series, we have that the result is the same for any order of the terms. As Wikipedia states:

"Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly."

It is also worth pointing out that similar phenomena occur in the context of integration; namely that integrals of functions whose absolute value can be integrated are in general much more useful.

(In particular I am referring to improper Riemann integrals, since integrals of this type are not even defined for the Lebesgue integral.)

If we work with conditionally convergent integrals, then we have to deal with similar problems as with conditionally convergent series; namely how we take the limit will affect the final result of the "integral". See, for example: https://en.wikipedia.org/wiki/Cauchy_principal_value

All of these problems are related to the fact that any consistent definition of

$$\infty - \infty$$

which is independent of the order of terms is impossible.

Proof:

Set $\infty - \infty = b$, $ b\in \mathbb{R}$. Then:

$$(b - \infty) + \infty = -\infty + \infty = -(\infty - \infty) = b$$

$$b + (-\infty + \infty)= b + -(\infty - \infty) = b-b=0$$

In other words, when we try to define $\infty - \infty$, then we completely lose any notion of associativity of addition -- this leads to all of the problems with rearrangement that occur with conditionally convergent series and integrals. (Suddenly order matters where it didn't before for finite sums.)

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    $\begingroup$ So it would be safe to say the notion of "time"/order is necessary in order to define a conditionally convergent series? $\endgroup$ – drewdles May 21 '16 at 13:35
  • $\begingroup$ Definitely, yeah. Even for Abel/Cesaro summation (to the best of my knowledge) the order will still matter -- their purpose is to define limits for series which are not even conditionally convergent (for example the harmonic series). $\endgroup$ – Chill2Macht May 21 '16 at 13:40

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