Series and Consistency Sometime ago I asked myself this question:


  Why manipulating $\sum_{n=1}^{\infty}\frac{1}{2^n}$ as $1$ leads to no contradiction in the arithmetic of $\mathbb{R}$ ?


More generally, why "treating" convergent series as numbers leads to no contradiction? I've found no satisfactory explanation to this question, maybe this is a meaningless question so nobody ever asked this.

I tried to come up with an answer, here it is:

We usually think of a convergent series as being a number, for example, $\sum_{n=1}^{\infty}\frac{1}{2^n}$ equals $1$.
But we should think the other way around, if $S$ is the set of all convergent series, we should think of "$=$" as being a function from $S$ to $\mathbb{R}$, that associates every convergent series to its limit. Of course, for every real number we can find a infinite series that converge to it, but there are many different series, so we need an equivalent relation on $S$, two series being equivalent if they have the same limit.

Is my answer ok?

Does my question even make sense?

Thanks.
I think you're misunderstanding what the notation "$\sum$" means. For an infinite summation, $$\sum_{i=1}^\infty a_i$$ is defined to be the unique real number $b$ such that the finite partial sums "approach $b$", or formally, such that$$\forall \epsilon&gt;0\exists N\forall m&gt;N\left(\left\vert b-\sum_{i=1}^ma_i\right\vert&lt;\epsilon\right),$$ if such a $b$ exists (note that at most one such $b$ exists).

We can now prove that e.g. $\sum_{i=1}^\infty2^{-i}=1$, using the basic axioms of the real numbers (and this is a good exercise). We can also prove that some summations don't correspond to any real number, e.g. $\sum_{i=1}^\infty 7$. 

Note that we're not allowed to just assume that an expression like "$\sum_{i=1}^\infty 2^{-i}$" corresponds to a real number (that is, is defined); we have to prove it. But once we've proved that it does, we've shown that "$\sum_{i=1}^\infty 2^{-i}$" is a name of some real number. Note that numbers have many names - e.g. $1=\sum_{i=1}^\infty 2^{-i}={7\over 7}=\sin{\pi\over 2}$. There's no problem with having multiple series converge to the same value, any more than there is with multiple fractions evaluating to the same number.



That said, the picture you've given of a function from [names] to [values] is a very important one, and will come up in many fields of mathematics (some keywords include free groups/rings/etc., groups/rings/etc. generated by some relations, and - in logic and universal algebra - term models). However, I'd argue that it's not really the right picture for this particular context.
One cannot always manipulate series is such a manner, even (conditionally) converging series.


  The Riemann series theorem says "if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges."


A famous one is the paradox shown on the Simpson's.  I will leave it to the reader to make conclusions about the manipulation of series.

You may find other examples here.