Flawed alternatives
It is known that the harmonic series 1 + 2 1 + 3 1 + … diverges. What about the alternating harmonic series 1 − 2 1 + 3 1 − … ?
Here, I will give a proof that the alternating harmonic series converges to 0!
Step 1: We rewrite the sum as follows.
1 − 2 1 + 3 1 − … = ( 1 + 2 1 + 3 1 + … ) − 2 ( 2 1 + 4 1 + 6 1 + … )
Step 2: We express the two separated sums in sigma notation.
( 1 + 2 1 + 3 1 + … ) − 2 ( 2 1 + 4 1 + 6 1 + … ) = ( n = 1 ∑ ∞ n 1 ) − 2 ( n = 1 ∑ ∞ 2 n 1 )
Step 3: We use the distributive property.
( n = 1 ∑ ∞ n 1 ) − 2 ( n = 1 ∑ ∞ 2 n 1 ) = ( n = 1 ∑ ∞ n 1 ) − ( n = 1 ∑ ∞ 2 n 2 ) = ( n = 1 ∑ ∞ n 1 ) − ( n = 1 ∑ ∞ n 1 )
Step 4 : We combine the two sums.
( n = 1 ∑ ∞ n 1 ) − ( n = 1 ∑ ∞ n 1 ) = n = 1 ∑ ∞ ( n 1 − n 1 ) = n = 1 ∑ ∞ 0 = 0
Which step(s) is/are flawed, causing this proof to be invalid?
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1 solution
My take was that, since it is invalid to rearrange terms of a series that is not absolutely convergent, step 1 is flawed. (While the alternating harmonic series is convergent, it is not absolutely convergent, (as the harmonic series diverges), and hence falls prey to the Riemann series theorem .) And as you say, we (generally) cannot combine (or separate) two divergent series, as we would be dealing with an indeterminate form of ∞ − ∞ .
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I've seen that theorem exploited in a Numberphile video, but I didn't know that's what the theorem was called. Thanks!
Never heard of that theorem! I'd never have imagined that permuting a conditionally convergent series would give you any real number; I would think that it would be maybe a countably infinite set.
The moral here - or at least, my moral here - is that " … " is very misleading. The rearrangement would be legal if the first … was taken as "to 2 N " and the second as "to N ".
This is slowly becoming one of my favourite problems, the more I think about it.
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Yeah, not all "..."'s are created equal. :)
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@Brian Charlesworth – All "..."'s are equal, but some "..."'s are more equal than others.
Sorry. Couldn't resist the Animal Farm reference.
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@Siddhartha Srivastava – Haha. Yeah, I should have seen that one coming. :)
"Also, it is wrong to combine two divergent sums". Can you give an example where the difference between two identical divergent sums is not 0 ? That was the only reason why I didn't pick "1 & 4", even though I agree that it's poor policy to try combining divergent sums in that fashion in general. Even a busted clocked is correct twice a day.
Next, maybe you should write a wiki why 1 + 2 + 3 + 4 + . . . = − 1 / 1 2 and explain how that is done without violating this same rule.
Edit: If you write
n = 1 ∑ ∞ n 1 − n = 1 ∑ ∞ n 1 = = n = 1 ∑ ∞ ( n 1 − n 1 )
Wolfram will return a value of T r u e
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That's interesting. How I looked at it was that "pairing" the merged series as done in step 4 was only one possible arrangement, and that we could form other arrangements that can lead to different results. Given that inconsistency, I reasoned that step 4 was invalid.
I also reasoned that, since we could not claim that, (or at least I wouldn't),
n = 1 ∑ ∞ n 1 − n = 1 ∑ ∞ n + 1 1 = n = 1 ∑ ∞ ( n 1 − n + 1 1 ) ,
as the left side is of indeterminate form and the right side is a telescoping series with a sum of 1 , (or, in the form n 2 + n 1 can be seen to converge by the p-test), it would be "bad form" to consider step 4 as valid.
That said, I haven't reshared this problem yet because I still have the usual niggling doubt that arises when dealing with infinite series.
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As I already pointed out, it's bad form to algebraically combine divergent sums. However, in the context of this problem, was step 4 indeed "incorrect"? Let's call out the thousand-dollar-an-hour lawyers.
This is one of the unfortunate things about mathematics. We want to believe that it should be free from such ambiguity, but there it is. It happens. During the 18th century, it was developing fast and furious, but by the 19th century, mathematicians began to worry and obsess about these kinds of ambiguities (shades of Nash paranoia, huh?), and finally it collapsed in a kerfuffle between Russell and Whitehead's Principia Mathematica and Godel's Incompleteness Theorem . Purist mathematicians still wring their hands and worry about it, but confidence has been badly shaken that mathematics could ever really be free of these irksome kinds of things.
Really, Jake Lai should be compelled to do a wiki on how the infinite sum of positive integers is -1/12. You know about that one, don't you?
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@Michael Mendrin – Yes, that one is notorious. My understanding is that it can be useful in certain instances to manufacture ways in which to "assign" finite values to divergent series, and that zeta function regularization and Ramanujan summation achieve this goal for this particular series. I find the statement 1 + 2 + 3 + . . . . = − 1 2 1 an abuse of our "usual" understanding of equality, but when it comes to infinite sums I suppose the concept of "usual" gets somewhat nebulous.
And that's just the start of it. There are so many ways of assigning values to divergent series it makes my head spin. But if they all have their uses then I guess it's best to embrace the weirdness and throw purity to the wolves.
As for the status of step 4, well .... Rather than lining the lawyer's pockets, perhaps those who answer either "1" or "1 & 4" should both get credit, since there is a reasonable doubt about 4's guilt. That doesn't mean that it's innocent, just that a jury of 12 mathematicians would end up deadlocked.
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@Brian Charlesworth – See my response to Jake. But mathematics is full of forks in the road, and whenever we take one path over another, we are simply axiomaticizing . Sometimes those choices are profound, other times it seems more like convention. The idea that an infinite sum of positive integers could be − 1 / 1 2 depends on a lot of those paths taken. But, as in quantum physics, mathematics is really about all paths taken. In fact, some of the most important discoveries in theoretical physics are a result of those less traveled mathematical paths taken. For instance, there's an amazing connection between the factoid that an infinite sum of positive integers being − 1 / 1 2 and the Casimir Effect in quantum field theory.
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@Michael Mendrin – Yes, I read about that connection while reading about the − 1 / 1 2 sum. I sometimes wonder if every mathematical result, no matter how abstract or obscure, will eventually be found to have a connection with some entity or process in our, (or some other), universe, (even if it's millennia down the road.)
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@Brian Charlesworth – Physics has always followed this course: "We're almost done with physics! We just need to patch up this little phenomenon, then it'll be complete."
Of course, we know how it goes: those little ϵ 's spawned relativity and QM and string theory. That could be a heuristic argument as to why every mathematical result which eventually find connection to physical reality.
I don't know, I'm spouting nonsense here. I'm down with a bad cold and a sore throat.
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@Jake Lai – Sorry you're sick. :( Get well soon.
Well, if the universe lasts forever then everything that can possibly happen will happen, and every mathematical result that can be achieved will be achieved. These two sets will then necessarily form a one-to-one correspondence, proving that reality is nothing more than the realization of mathematics, i.e., God is a mathematician. :D
@Michael Mendrin – It's generally incorrect to so haphazardly throw together two not-absolutely-convergent sums and call them equal.
I should, but I'm not versed enough in complex analysis and analytic continuation to do so.
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@Jake Lai – Jake, I am actually going with your interpretation of this problem, which is that as a general policy one shouldn't even be thinking about doing algebraic manipulations of divergent sums, but I thought maybe this was some kind of a gotcha question. Even Wolfram agrees with me in this instance, but I'm agreeing with you nonetheless, so no change needed.
Dear Michael Mendrin,
The young people are correct for this particular question, namely Step 4 is a flaw!
After an analysis, I started to realize the fact that two different infinities when combined into one infinity makes a fault to diverging series because a result of value approximated to Ln (infinity/ Huge) or Ln (Huge/ Infinity) could be resulted, which is not zero indeed! Only the R.H.S. is zero and hence cannot be equated.
I made the same mistake. We just can't simply combine two diverging series just like that when we haven't been sure about the extend of each infinity! This seems illogical and cruel but TRUE!
I have finally agreed and concluded that you are correct! The answer is absolutely correct.
The sum of 1/ n approximated to Ln (infinity/ huge) for example which is an indeterminate. We cannot combine two diverging series into one which ignores different extends compared to the only extend! I apologize for my previous faults.
Congratulation for being correct!
The key thing to note is that the sum is actually
N → ∞ lim ( n = 1 ∑ 2 N n 1 ) − 2 ( n = 1 ∑ N 2 n 1 )
The fallacy is that in this flawed proof, it is taken that 2 N = N , ie the sum is
N → ∞ lim ( n = 1 ∑ N n 1 ) − 2 ( n = 1 ∑ N 2 n 1 )
Also, it is wrong to combine two divergent sums.