Explain the 1 + 2 + 3 in $ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} $ $$ \frac{1 + 1 + 1 + \cdots}{1 + 2 + 3 + \cdots} = \lim_{n \to \infty} \frac{1}{(n+1)/2} = 0 $$
If $n$ goes to infinity, we can image that a bit by taking a very big number. 
Like $1.000.000.000$
$1.000.000.000+1=1.000.000.001$
$1.000.000.001/2=500.000.000,5$
Now how does the $1 + 2 + 3 + ...$ make sense?
 A: They're using that $$1+2+3+\dots+n=\frac{n(n+1)}2$$
For example $1+2+3=6=\dfrac{3\cdot 4}{2}$
On the other hand $$\underbrace{1+1+1+\cdots+1}_{n \;\;\rm times}=n$$ 
Thus $$\frac{1+1+1+\cdots}{1+2+3+\cdots}=\frac{n}{\dfrac{n(n+1)}2}=\frac{2}{n+1}$$
Of course, we're interpreting $$\frac{1+1+1+\cdots}{1+2+3+\cdots}$$ as $$\lim\limits_{n\to\infty}\frac{\sum_{i=1}^n 1}{\sum_{i=1}^n i}$$
If we had, for example $$\lim\limits_{n\to\infty}\frac{\sum_{i=1}^{n^2} 1}{\sum_{i=1}^n i}$$
then we would say that $$\frac{1+1+1+\cdots}{1+2+3+\cdots}=2$$
We can write this more suggestively as $$\frac{{1 + (1 + 1 + 1) + (1 + 1 + 1 + 1 + 1) +  \cdots }}{{1 + 2 + 3 +  \cdots }} = 2$$ to make clear how many terms we add in each step.
The moral of the story is that you cannot have "voids" in your notation. Writing something like $$\frac{1+1+1+\dots}{1+2+3+\dots}=\text{something}$$
should be better replaced by something informative and clear that really says what is going on.
A: Look user, you are brining up a great issue here! The numerical expression itself is pretty meaningless if we do not know the context, i.e. the formula where this expression came from. With regrouping symbols I can get virtually anything out of the given numerical expression, because, as stated, it is an infinity / infinity situation, which is indeterminate. The word "indeterminate" is really appropriate here. Sure, the answer could be 1/2 if the problem arises from n/2n, but with grouping symbols, the problem also could arive from 2n/2n in which the answer would be 1. If I group enough 1's together, I can also get 4 as an answer. As long as we do not know from which expression your numerical problem comes from, there is no point arguing what answer it has to be.
A: It doesn't in general make sense to take ratios of infinite series because convergence may not be assured. In this case, the numerator and denominator diverge so you have to be very careful about what you mean by
$$\frac{1+1+1+\cdots}{1+2+3+\cdots}.$$
The best way to interpret this ratio is as the following:
$$\frac{1+1+1+\cdots}{1+2+3+\cdots} = \lim_{n\rightarrow\infty}\frac{\sum_{i=1}^n1}{\sum_{i=1}^ni} = \lim_{n\rightarrow\infty} \frac{n}{\frac{n(n+1)}{2}} = \lim_{n\rightarrow\infty}\frac{2}{n+1} = 0.$$
