When one goes to add two arithmetic series together, I’ve often seen the practice of “bumping” one over by one place, which gives a different result from just adding them together.
For instance, the proof of S=1-1+1-1+1-… = 1/2 goes like this:
S=1-1+1-1+1-1+…
+S= 1-1+1-1+1-…
2S = 1
S=1/2
My google fu is weak, and I can’t find the reason for why it’s proper and holy to do the shift of the series in the second line, when it creates a different result than if there was no shift in the second line.
The short answer is because mathematicians have found that allowing such produces interesting and unambiguous results, that show consistency with other interesting and unambiguous methods. Math is, fundamentally, a game, to which we make up the rules, and we can make up whatever rules we want.
It is not allowed (in general). Infinite series have to have certain properties in order to carry out certain operations.
The series you give does not have any reasonable properties so the operation you are trying to do doesn’t work.
It’s like division. You can divide a given real number by any real number that has certain properties. Well, one property: the divisor can’t be zero.
The big property an infinite series needs to be able to apply basic operations is convergence. But even non-convergent series can be operated on depending on how you definte their summation. E.g., Cesàro summation.
Note that there is a lot of background deep Math behind the rules of all this that doesn’t look at all like the operations in the OP. But this background is too complex for day-to-day use so you first show you can do the simpler version.
ftg, I started with your link on Cesàro summation, through to Grandi’s series, and followed through with others from there. That helped clarify a few things. Thank you!
The thing to remember about infinite series is that the associative property of addition does not hold for them. So the order that you sum them matters. In fact any conditionally convergent series can be rearranged such that the sum can produce any number.
So for your original question, the reason why it’s ok to rearrange the series like that is that nothing expressly forbids it. But you should not be surprised if you see different results depending on how you rearrange it. Similar to how (23) + 1 is different to 2(3+1) is a valid rearrangement but will produce different results.
Exactly. I think it was early high school math. Addition and subtraction are commutative so a+b=b+a, you can rearrange a list of numbers in addition and subtraction any way you want.
The “gotcha” is that it works only for finite series. Once things become infinite, there is no precise answer.
After all, 1+1-1+1-1+1-1+1… = 1-1+1-1+1-1+1… = -1+1-1+1-1…
you can play any trick you want like that.
An infinite series does not have a “running total” the only relevant number is the final total - it either converges or it does not. If it does not converge, then the result of any manipulation like you show is meaningless.
I disagree. The running total is called the partial sum and the definition of a convergent infinite series is that the sequence produced by the partial sums converges.
To be clear rearrangement of divergent series will not produce any different results. You can sum up a divergent series however you like and it will still diverge. This is also true for absolutely convergent series. No matter how you add up an absolutely convergent series you will get the same result.
It’s actually only conditionally convergent series where rearrangement of the elements can produce a different result(IE the associative property doesn’t hold). A conditionally convergent series must have infinite positive and negative terms. Thus you can rearrange these to produce any real number you’d like.
Knowing that you can get any result that you’d like out of your series if you rearrange it should suggest that you should be very careful when you decide to rearrange things.
I would say the “simple version” is that the given series does not converge, which fact is obvious, therefore the manipulation, not to mention the conclusion, in the OP is erroneous. Notwithstanding further analysis that can give rise to strange-looking formulas like 1 - 1 + 1 - 1 + … = 1/2, 1 + 1 + … = -1/2 (which, if you add them, seem to inconsistently yield 2 + 0 + 2 + 0 + … = 0, so at that point you had better understand what you are doing…)
Correction to what I said earlier. When I said associative property I meant communicative property. I get these mixed up all the time.
I’m actually not sure about rearrangements of divergent series. I originally just thought about it and assumed it didn’t matter. But I didn’t have a good reason to believe rearrangements of divergent series would always be divergent. It seems it’s much more complicated than I originally thought.
A conditionally-convergent series that’s arranged in the “wrong” order is a divergent series. So yes, it’s absolutely possible (in some cases) to re-arrange a divergent series so that it converges.
But the point is that the limit of the partial sums is not the only possible way to define the “sum” of an infinite series. Under that definition, the series 1 - 1 + 1 - 1 … does not have a sum. But there are other definitions under which it does have a sum.
Again, it’s just a different set of rules for a game. It’s like asking “How is it that a baseball team is allowed to replace the pitcher with a strong slugger, and then put the pitcher back in in the next half-inning?”. To which one could answer “They’re not allowed to do that”. But one could also answer “They’re an AL team, and they have the designated hitter rule”. It’s still baseball.
^ Ironically enough, I was going to show someone the Numberphile proof of that, which is what motivated my question in the first place! He uses that shifting move on the 1-2+3-4+… series.
But it’s not wrong. It’s just a new definition. You find those in math all the friggin’ time. Before you even get to infinite series, you’ve already re-defined addition at least a half-dozen times, and even more re-definitions of other operations.