MasterOfDisaster

Hey, I don't even know where to post this, maybe it could be moved to where it belongs.
So when I look at very complex mathematical forumals, they look like gibberish to me, and I'm quite happy to find that they are gibberish, but this seems to quite directly conflict with my logic.. so I'm not asking to be explained this in a way that I understand, I would be more than happy if someone could translate this, which in my head right now is a fatal critical error contradiction, into just regular gibberish.. or something like that.. tell me it's something I must be misunderstanding.. or something lol

http://en.wikipedia.org/wiki/1_%2B_2_%2 ... _%E2%80%A6
This is the wiki article about it.

MarkFL

where the notation indicates Ramanujan summation.

http://en.wikipedia.org/wiki/Ramanujan_summation

I moved this to Real Analysis & Topology.

johnny

Assuming no advanced mathematics knowledge, wouldn't 1 + 2 + 3 + ... diverge to infinity?

The Chaz

Or it converges to some p-adic number, perhaps

MasterOfDisaster

Okay, I think I'll leave this as it is for now, I was showed it by someone who just wanted to annoy me xD

manus

S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = 1/2
this is not true 1-1+1-1....=0
or 1-1+1-1....=1
the outcomes can not be mediated.

On the other hand true is .....99999.0 = -1
so 9+90+900+..... = -1
and 1+10+100.....=-1/9

v8archie

It very much depends on what you mean by your notation. The sequence of finite sums does not converge, but there is a meaningful sense in which we can attach the value $-{1 \over 12}$

What does this mean?

manus

Informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are quite close to one another; and if they differ only after the 20th decimal place, they are even closer.

10-adic numbers use a similar non-terminating expansion, but with a different concept of "closeness". Whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 3333 and 4333, which differ by 10^3, are close in the 10-adic world, and 33333333 and 43333333 are even closer, differing by 10 ^7.

More precisely, a positive rational number r can be expressed as 10^e·(p/q), where p and q are positive integers and q is relatively prime to p and to 10. For each r ≠ 0 there exists the maximal e such that this representation is possible. Let the 10-adic absolute value of r to be

|r|= 10^-e
|0| = 0.
Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers x and y is given by |x − y|. An interesting consequence of the 10-adic metric (or of a p-adic metric) is that there is no longer a need for the negative sign. As an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number −1:

9=-1+10 so |9-(-1)|= 0.1
99=-1+10^2, so |99-(-1)|=0.01
999=-1+10^3 so |999-(-1)|= 0.001
9999=-1+10^4 so |9999-(-1)|=0.0001
and taking this sequence to its limit, we can say that the 10-adic expansion of −1 is

...... 9999=-1

manus

5-adic:
-1 = .......444
i = .......12013233
-i = ......32431212

v8archie

That's a completely different number system, so it's rather muddying the waters here.