0 + 0 + ... = ?

Interestingly the link of this discussion is lichess.org/forum/off-topic-discussion/0--0----

I think the usage of "..." is too unspecific in this audience
Nevertheless, one could have

These limits are the same because every element of the limits are the same:
a = 0 + 0 + ...
a/2 = 0/2 + 0/2 + ... = 0 + 0 + ...

The limit is a real number between -1 and 1, because the sequences
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
-1/2 + -1/4 + -1/8 + -1/16 + ... = -1
Provide an upper and lower bound

So now
a = a/2
a - a/2 = 0
a/2 = 0
a = 0

> As an afterthought: this is also what @m011235 got wrong in #13:
> > Limits of the form 0 * infinity are sometimes defined
>
> No. Limits in the form of n APPROACHING 0 are defined, but that means that said n is always an epsilon away from 0.

You should have quoted the whole phrase:

> > Limits of the form 0 * infinity are sometimes defined, for example take n -> + inf and 1/n -> 0, but n * 1/n -> 1.

I provided an example precisely illustrating what you wrongly affirm is missing, so the alert reader can infer that by herself.

Himself*
By the way I am a noob in math :(
And I'm just telling what was taught in school...

@kyanite111 I had the same opinion on the age of many posting on here. But I have now decided that I had overegged the pudding. It’s actually more like 3. On a good day.

@TakeThePawnOrLose said in #4:
> However, infinity can be rewritten as the limit(x->0) n/x where n is any number. Using this, we get:
> limit(x->0) x(n/x) = n.

This limit, in fact, does not exist for any real number n.

@m011235 said in #23:
> You should have quoted the whole phrase:
> > Limits of the form 0 * infinity are sometimes defined, for example take n -> + inf and 1/n -> 0, but n * 1/n -> 1.

Again: no. What you write here is: "n/n" and that equals always 1.

> n * 1/n -> 1

n * 1/n *==* 1, not "-> 1". It doesn't matter what "n" is, n/n (n * 1/n is just writing it in another form) is always 1 and nothing else. Otherwise something would be wrong with division and/or multiplication.

Limits are different: For n->inf lim (1/n) -> 0, but: the bigger n gets, the closer you get to 0, but you will always be an epsilon away from it. This is why you can multiply lim(1,n) by n and get any number (even approaching infinity) but

> <anything, including infinity> times 0

is still 0. Conversely, dividing x/lim(y ->0) will get you a result (in fact it approaches infinity), but x/0 is still undefined. The reason is the difference between something *approaching* zero and *equalling* zero.

@Nomen-Nonatur Of course 1/n * n = 1, this is the whole point, using a simple and concise example. If equality bothers you, just replace by (1/n) * (n+1) -> 1, and voilà! We have a limit of the form 0 * inf = 1. Hint: 0 is 1/n, inf is n+1. Which was my very statement:

> Limits of the form 0 * infinity are sometimes defined

@TakeThePawnOrLose said in #4:
> Proof:
> 0+0+0+... = 0*inf

Wrooong. Factorization is only defined for real numbers. Infinity is not a real number.

My wallet balance + House = 0