Formally, I understand that infinite series are not defined by adding up "infinitely many" terms, but are instead defined as equalling their limit. As user Brian M. Scott outlined in an answer to a similar question (Why is an infinite series not considered an infinite sum of terms?), it is easier to define infinite series by the limit of their partial sums than by considering every single term. However, one geometric proof of the convergence of $1/2 + 1/4 +1/8+1/16+ \cdots$ has made me question why we circumvent the problem of adding infinitely many terms with the concept of the limit:
Image credit: https://www.mathsisfun.com/algebra/infinite-series.html
When you look at the diagram above, it seems like every single term has been included (not literally, but it is clear what the diagram represents). Furthermore, if you plotted the above shape on the Cartesian plane, the area of the shape would be $1$. Even the point $(0.99999,0.99999)$ would be covered by a square/rectangle. When all of the terms have been plotted, it seems like the infinite series not only tends to 1, it equals $1$. To me, this is not just because we define an infinite series to equal its limit. The limit only concerns the partial sums, whereas the diagram shows that even if we allow ourselves to add infinitely many terms, there is no immediate contradiction. Obviously, defining infinite series like this formally can create problems: infinite series do not have always have the commutative property, for example. However, is there anything conceptually wrong with thinking of infinite series as adding up infinitely many terms, even though technically this can lead to some problems?
Best Answer
The answer to your question is essentially here:
It is intuitively OK to think of summing an infinite series as
but if you want to prove theorems about those sums and avoid the problems you need a formal definition. Mathematicians have discovered that a good way to do that is to avoid "adding infinitely many numbers" by proving infinitely many statements. That's what the "for every $\epsilon$" does in the formal definition of limits.