Why does the concept of limits necessitate that infinite series can be assigned a value?

When summing an infinite series such as $1/2+1/4+1/8+1/16+...$ I have often seen the following sort of argument:

• The partial sums of the series cannot equal more than 1, as when you add another term the distance between the partial sums and 1 halves (e.g. before you added $1/8$ to the series, the difference between the partial sum and 1 is $1/4$, now it is 1/8). This shows that however many terms you add, the partial sums are always below than 1



• The difference between the partial sums and 1 is smaller than any positive number (i.e. if I pick an arbitrarily small number such as 0.00005, at some point the difference between the partial sums and 1 will be less than this)


• A number smaller than any positive number is known as an infinitesimal


• There are no non-zero infinitesimals (Archimedean property). Hence, the infinite sum must equal 1 exactly

This argument shows that if the limit of a series is 1 then the series must equal 1. However, can't I use the same kind of argument to show $1/0$ = $\infty$ (which is clearly false)?

• As $x$ approaches 0 from the positive end, $1/x$ approaches $\infty$


• Therefore, the limit of $1/x$ as $x$ approaches 0 is $\infty$



• So 1/0 = $\infty$

Aside from the problem that approaching 0 from the negative end yields -$\infty$, there seems to be a deeper underlying problem with my reasoning. What is this? (If possible a non-technical answer would be greatly appreciated.)

Answer

If some series approaches $\infty$, it is pretty standard to say that series has no limit, for this exact reason. The limit is unbounded. $\infty$ does not exist, so it makes little sense to say it actually equals infinity. I've written before that

$$\lim_{x\to0}\frac{1}{x}=\infty$$

but this is more of a shorthand or even an aesthetical choice than it is appropriate. It is really understood to say that the limit does not exist at all, because there is no limit.

To comment on one of your follow up questions in the comments where you say

So the principle that if the limit of something is $x$, then that something must equal $x$ only applies to infinite series?

The answer really depends. When we discuss discontinuous functions with a single point removed at $x=a$, you can have $\lim_{x\to a} f(x)=b$ both from the left and from the right, but $f(a)=c\neq b$, so in this case the principle that if the limit is $b$ then that something must equal $b$ is incorrect.