The numerical relation of the sum of two divergence series For these two series:
$1 + 2 + 3 + 4 + 5 +...$
$2 + 4 + 6 + 8 + 10 +...$
For each of the two series, since these numbers progress with no end, and the sum increases,
it certainly cannot be finite. By this fact it becomes infinite. Hence, this quantity is so large
that it is greater than any finite quantity. Thus I think the sum could only be an "infinite number", which is
greater than any finite or assignable quantity. Moreover, every term of the second series is
twice as large as the corresponding term of the first series(1 to 2, and 2 to 4, and n to 2n), it is reasonable to believe that the
sum of the second series is twice as large as the first series, i.e. we get two infinite numbers
and one is twice as large as the other.
so is there something wrong here?
 A: I don't know if i am wrong, but couldn' we simply say
$$S_n = 1+2+...+n=\frac{n(n+1)}{2}$$ and $$P_n=2(1+2+3+...+n)=2\frac{n(n+1)}{2}$$.
If we look at the ratio 
$$q_n=\frac{S_n}{P_n}=\frac{1}{2}\frac{S_n}{S_n}=\frac{1}{2}$$
Hence, couldn't we conclude ?
$$q = \lim_{n \to \infty}q_n=\frac{1}{2}$$
So it seems to be quite obvious that even for $n \to \infty$ one could say, that the second series is twice as large as the first series.
A: What I think is wrong with your reasoning is the perception of infinity as a definable quantity (the statement "one infinity is twice of another" assumes two different infinities to be defined quantitatively. Equivalent statement would be that the two infinities are identifiable as specific points in the cartesian co-ordinate space). That is leading to your confusion.
Here are a few statements that would help you put things in perspective -


*

*The two series grow indefinitely and 'get lost in the clouds'. Mathematicians are usually lazy enough to avoid writing this in so many words. They devised this nice symbol $\infty$ for this purpose.

*The only quantitative comparisons that hold meaning involve measurable quantities, which by implication, must be finite. Hence, the moment you start thinking about the word 'twice', there's no point 'worrying what's above the clouds'.

*Note that though the two series are infinite, they are completely defined using set builder definition thus: $$ S_1=\{x_n|x_{n+1}=x_n+1, x_1=1, n \in \Bbb N\} $$ $$ S_2=\{x_n|x_{n+1}=x_n+2, x_1=2, n \in \Bbb N\} $$
This is useful to realize that though we cannot make meaningful statements about the sum of all elements in $S_1$ and $S_2$ as measurable quantities, we can draw conclusions about their behaviour based on the observed patterns of their rising partial sums till the $t$'th term $$ S_{1,t}=\{x_n|x_{n+1}=x_n+1, x_1=1, n \in \Bbb N,  n \le t\} $$ $$ S_{2,t}=\{x_n|x_{n+1}=x_n+2, x_1=2, n \in \Bbb N, n \le t\} $$
It is easy to see that sums of all elements in $S_{1,t}$ and $S_{2,t}$ are finite as long as $t$ is finite. And by using induction we find that $$\sum S_{2,t} =2* \sum S_{1,t} , \forall t \in \Bbb N$$
which is a meaningful quantitative comparison between the two series that gives a sense of 'HOW these two series are headed in the clouds'. 


Note that definite conclusion could be drawn about the behaviour of the two series even though they are infinite because their definition is fully capture-able given any one element of the series and a set of well-defined mathematical operations that could be used to generate other elements
The word 'twice', and the quantitative comparison it implies, belongs to the domain of partial sums (these are finite) alone, and NOT to the convenience called $\infty$
