Is the sum $2+2+2+2+\dotsb$ even?

Question

I have failed to find a relevant question, so I am posting one. As the question says, is $2+2+2+\dotsb$ even or odd? Or since this "number" is not an integer, it might be undefined.

Furthermore, are the following two expressions equivalent?

$$1+1+1+\dotsb = 1+(1+1)+(1+1+1)+\dotsb$$

Now "$+$" is an associative binary operation, but I have a scratchy feeling they should not be the same because of the idea of infinity. Lastly, I want to know more about (absolutely or conditionally) divergent series, and is there a good reference I can look into? Thank you in advance.

Answer 1

The notation $$ 2+2+2+2+\cdots $$ does not have a commonly accepted meaning, so it is fruitless to ask whether the number it stands for is odd or even: It does not stand for any number at all.

By its form, the notation would seem to be the infinite series $$ \sum_{k=1}^\infty 2 $$ but this series does not converge, and therefore is not considered to have any number (odd or even) as its value.

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There are some summation methods for divergent series that propose values such an expression might be considered to have; for this series the most common answer would be $-1$, which is odd. But usually these summation methods are not taken to be the meaning of an expression such as the above unless that is explicitly said in the context.

Answer 2

Suppose you have sequence of positive integers, $\,a_1,a_2,a_3,\dots,\,$ then the sequence of partial sums $\,a_1,\,a_1+a_2,\,a_1+a_2+a_3,\dots\,$ diverges to infinity. In other words, $\,a_1 + a_2 + a_3 + \dots\,$ is not a convergent series and so it does not have a sum and has no value, although each of its parital sums is a positive integer. In some sense, all such series are equivalent since they all diverge to infinity.

Answer 3

Your question doesn't really make sense since 2+2+2+2+2+... isn't a number in the commonly accepted sense. But, as ViHart says, math is all about making stuff up and seeing what happens, so it could be fun to try to think about an answer to your question anyway.

What does 2+2+2+2+... mean? What does it mean to you? Some people may see it as a divergent series, but we could also see it as a sequence of integers. What could it mean for an arbitrary sequence of integers to be "even" or "odd"? Maybe a sequence of only even numbers is even, and maybe a sequence with only a finite odd number of odd numbers is odd, and otherwise it's undefined.

Anyway, there is no standard definition of "odd" or "even" for things like 2+2+2+2+..., but that shouldn't stop you from trying to come up with a meaning yourself! Maybe it could lead to some fun and interesting math. Just think of Ramanujan giving meaning to seemingly divergent summations!

Answer 4

Since nobody has touched the second question, I'll add that for series of non-negative numbers, changes in grouping (and even order) do not affect the value of the series. For instance, $$1+1+\ldots$$ and $$1+(1+1)+(1+1+1)\ldots=1+2+3\ldots$$ both diverge to infinity (and thus don't contradict this principle).

However, once you allow negative numbers, it is no longer true. The classic example is $1-1+1-1\ldots$ diverges, while $(1-1)+(1-1)+\ldots$ converges to zero.

(On the other hand you are using an alternate version of summation like Henning mentions in his answer, where divergent series can take numerical values, it might no longer be the case that $1+1+\ldots$ and $1+2+3+\ldots$ are the same. For instance, their Ramanujan sums are $-1/2$ and $-1/12$ respectively.)

Answer 5

I may not be right but...

$$x=2+2+2+...=a$$ $$x-2=2+2+2+...=a$$

This has no solution.