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2 solutions
Grandi's Series.
Any Divergent Series. (Grandi's Series is also an divergent series.)
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Yes but that might not be the best example since it does have exactly one Cesàro sum, namely
1
/
2
.
I cannot think of a better example at this late hour though. Are there any series that can be rearranged to more than just
1
sum? I was thinking maybe
ζ
(
0
)
but once again that has only one sum, assuming the initial restraint
n
=
1
.
I have made far too many mistakes tonight so I think it's time for sleep, but I'll be back tomorrow.
By the way, I made the mistake of pairing
r
with
1
/
r
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Grandi's series can easily be rearranged to obtain many different sums.
In fact, any sequence that is not absolutely convergent, and whose sum of positive terms diverge to infinity + sum of negative terms diverge to negative infinity, will allow you to rearrange them in some sequence so as to get a different summation.
Let S = ∑ i = 0 ∞ ( − 1 ) i = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + ⋯
Therefore, S = 1 + ( − 1 + 1 ) + ( − 1 + 1 ) + ( − 1 + 1 ) + ⋯ = 1 + 0 + 0 + ⋯ = 1
But,
S = ( 1 − 1 ) + ( 1 − 1 ) + ( 1 − 1 ) + ( 1 − 1 ) + ⋯ = 0 + 0 + 0 + ⋯
But, in fact, this is a geometric progression with a ratio -1.
Hence, S = 2 1 , the accepted answer.
Just, because an 'infinite' series sum can be computed in more than one manner, does not necessarily mean the sum is undefined.
The accepted result could be more absurd. (Eg : Sum of integers being a fraction, sum of positive integers being less than zero etc.,)
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One important distinction between what I wrote, and the summation that you wrote, is that I did not define the order of the product. This introduces a lot more issues, because, as I pointed out, by "arbitrary choice" of ordering, I can make the product into whatever I want it to be. For example, even with the Grandi sequence, if I take the 1st, 3rd, 2nd, 5th, 4th, 7th, 6th, .... terms, then, by your argument, the sum will be 1.5.
This is an extremely important fact. The order in which something is summed / producted, could affect the final product. This starts to enter the realm of analysis.
For summations, the final sum is invariant under changing the order of terms if the summation is absolutely convergent. Clearly, ∑ ∣ ( − 1 ) n ∣ is not convergent, which is why it (could) run into this issue.
For products, we can convert them into a summation by taking logarithms. We see the same thing happening here, where we want to try and determine ∑ lo g r . However, because this has infinitely many terms that tend off to positive and negative infinitely, this summation will not be able to converge at all.
An equivalent comparison would be to ask for ∫ − 1 1 x 1 d x . Many people say "oh, clearly the negative and positive parts will cancel out", but this is not true. Instead, this integral is undefined.
Grandi's series has an "alternative interpretation" using the idea of a Cesaro summation, which agrees for summations that already converge, and allows us to extend it out for some divergent scenarios like this.
However, note that Cesaro doesn't allow us to show that ∑ i = − 1 2 1 , which is another favorite summation that many people like to quote without really understanding the technical details of it. This requires using zeta function regularization.
As such, knowing the type of summation rules that you are working with is important, and there are different "convergence neighborhoods". With the lack of information, we generally assume that we're using the standard summation format.
Just a caviot: if I remember correctly page 24-26 in string theory volume 1 by Polchinski shows a proof in which the summation of all positive integers to infinity is equal to -1/12.
It is strange that nobody noticed that the argument that this is a geometric series which sums to 2 1 is clearly flawed. From the formula for the sum of geometric series we get the following expression: S = n → ∞ lim 2 1 − ( − 1 ) n , which clearly does not converge. The resoning which OP used works only when ∣ a n ∣ < 1 .
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The reason for nobody noticing is that the argument of 'sum of geometric progression, summing to ...' was not made. The series is a geometric progression with ratio -1. However, the sum is not arrived from the formula used for GPs.
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@Janardhanan Sivaramakrishnan – I am sorry for misinterpretation. Thank you for pointing it out.
Do we prove by the convergence and divergence test?
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Because the order of the sequence is not stated, if the limit exists then it has to be independent of the order used. Above, I had shown that the limit is not independent, hence the limit does not exist.
More generally, for a summation to be independent of the order used, a necessary and sufficient condition is for it to converge absolutely. For products, we first take logarithms, which converts them into a summation. We thus need the sum of the absolute value of the logarithm of all these terms, to be finite.
If one considers just every rational number more than 1 in this product, then the answer would be ∞ .
If one considers just the fractions between 0 and 1, the answer approaches 0.
When you multiply by something less than 1 and more than 0, your product decreases; When you multiply by something more than 1, the product increases; When you multiply by 1, your product stays the same. Calculating a product that is correct to a specified tolerance is impossible because generating this product depends upon chosen sequence of the factors of the infinite product.
[This is not a complete solution.]
Most people will be tempted to say that the answer is 1, because "you can pair up a number with it's reciprocal".
However, I can also choose a different pairing, say of r with 2 r 1 . If so, the product would be ( 2 1 ) ∞ = 0 .
Someone else might decide to choose a different pairing, say of 2 r with r 1 . If so, the product would be 2 ∞ = ∞ .
This tells us that the product is dependent on the order of terms chosen, and hence is an undefined product as stated.
How do we go about showing that this product is indeed undefined?
Hint: What kind of summation do you know has terms which could be rearranged to give us many different values?