Is it acceptable to say that a divergent series that tends to infinity is 'equal to' infinity?

Consider a divergent series that tends to infinity such as $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$. The limit of this series is unbounded, and I have often seen people say that the sum 'equals infinity' as a shorthand for this. However, is it acceptable to write $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots = \infty$ in formal mathematics, or is it better to denote that the limit is equal to infinity? If so, how does one do this?

Yes - it is both very common and entirely correct to do so. There is a bit of formal trickery here because $\infty$ is not a number, but you can do analysis with it anyways - meaning limits and that sort of thing. In particular, there is a set called the affinely extended reals which is basically the real numbers $\mathbb R$ along with two new objects $\infty$ and $-\infty$, one at each 'end'. This is a topological space, meaning that you can take limits in it, but be careful that some things like $∞-∞$, $0·∞$, $0/0$ and $∞/∞$ are undefined.

Consider that, for real numbers $x$, the definition of a sequence $s_n$ converging to $x$ is as follows:

For any $\varepsilon >0$, there exists some $N$ such that if $n>N$ then $|s_n-x|<\varepsilon$.

This can be rewritten as saying:

For any open interval $I$ containing $x$, there exists some $N$ such that for all $n > N$ we have $s_n\in I$.

The idea behind either of these definition is that if we choose some "neighborhood" of $x$ - consisting of $x$ and at least some positive radius around $x$ - the sequence eventually is constrained in that neighborhood. More formally, a neighborhood of a real number is any set $S$ containing an open interval around $x$. Then, you can define convergence to $x$ as follows:

For any neighborhood $I$ of $x$, there exists some $N$ such that for all $n >N $ we have $s_n\in I$.

To define limits to $\infty$ and $-\infty$, one just needs to define their neighborhoods. In particular, $\infty$ is meant to be the "upper end" of the real line - and being close to $\infty$ means that a number is very large. So one defines a neighborhood of $\infty$ to be any set $I$ containing an interval of the form $(C,\infty]$ for some $C\in\mathbb R$. Then, we say

$\lim_{n\rightarrow\infty} s_n = \infty$ if for every neighborhood $I$ of $\infty$, there exists some $N$ such that if $n>N$ then $s_n\in I$.

This is equivalent to saying that $s_n$ converges to $\infty$ if, for every $C$, there exists an $N$ such that if $n>N$ then $s_n > C$ - which is the usual definition you find in textbooks (but note that it is actually a theorem - a consequence of the definition of $\infty$!) - and that in any context that you might allow a statement like $\lim_{n\rightarrow\infty}s_n = \infty$, you might as well be working in the extended reals.

Then, since infinite sums are just limits of partial sums, it is perfectly rigorous to write $$\sum_{n\rightarrow\infty}\frac{1}n = \infty$$ and to know that this truly means that the left hand side evaluates to $\infty$, not to think that this is some special statement where equality is not equality. This is actually very common in real analysis (the branch of mathematics dealing with limits, continuity, differentiability, and all that stuff) - especially in subfields like measure theory and sometimes in the theory of metric spaces as well.

However, it is also important to know that many people do not share the view that $\infty$ is always a perfectly valid object, defined by its neighborhoods. So, even though you would technically be right to write such an equality, it might not go over well with your audience nonetheless - and you should keep your audience in mind whenever you write anything because "formal correctness" is no substitute for "understood by your audience" - and you will often encounter examples of things which are technically correct, but might confuse or annoy your audience nonetheless.

(Sidenote: The limit $n\rightarrow\infty$ in the subscript $\lim_{n\rightarrow\infty}s_n$, as you might notice, is also defined by the neighborhoods: we get to restrict $s_n$ by forcing $n$ to lie in some neighborhood of $\infty$ that we get to choose - which is what's going on when we say that there's some $N$ so that if $n>N$, blah blah blah)

The only thing that you can correctly write is that a limit is equal to infinity.

Be aware that writing this is nothing but a shortcut for the longer sentence $\forall M\in \Bbb R,\exists n\in \Bbb N, $ etc. It doesn't mean that there is a limit which is a number and that this number is $\infty$.

• Let $\{s_n\}$ be a sequence of real numbers with the following property:

  For every real $M$ there is an integer $N$ such that $n\geq N$ implies $s_n\geq M$.

  We then write $\displaystyle \lim_{n\to\infty}s_n=\infty. $ This expression has a clear definition as above. We sometimes write $+\infty$ instead of $\infty$.

• In your example, by the definition above, one writes $\displaystyle \sum_{k=1}^\infty\frac{1}{k}:=\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}=\infty. $

• In real analysis, one very often works with the extended real numbers, especially in the theory of measure and integration. One can interpret the symbol $\infty$ as an extended real number and the expression $\displaystyle\lim_{n\to\infty}s_n=\infty$ can be understood as convergence in $\mathbb{R}\cup\{\infty,-\infty\}$ with the order topology. One may also work with the extended non-negative real axis $[0,+\infty]$ with the extended topology. See for instance, this set of notes.

• Moreover, observe that a function $n \mapsto x_n$ from the extended natural numbers ${\Bbb N} \cup \{\infty\}$ (with the order topology) into a topological space $X$ is continuous if and only if $x_n \to x_{\infty}$ as $n \to \infty$, so one can interpret convergence of sequences as a special case of continuity.

"The limit of this series is unbounded"

Notes: "bounded/unbounded" is a concept for some subset of real numbers. One can say "some sequence of real numbers is unbounded", or "certain subset of real numbers is unbounded". It does not make sense to say that the limit of [the partial sums of] a series is "unbounded". In your particular case, one could say that the limit of [the partial sums of] the series "is not a real number" or "is not finite".

It is widely accepted for divergent series that tends to infinity the following short notation

$$\sum_{n=1}^\infty \frac1n =\infty$$

instead of the extended version usually introduced at first by the formal and rigorous definition

$$\lim_{N\to \infty }\sum_{n=1}^N \frac1n =\infty,\quad\sum_{n=1}^N \frac1n \to \infty \iff \forall M\in \mathbb R \quad\exists N_0\in \mathbb N \quad\forall N\ge N_0 \quad \sum_{n=1}^N \frac1n>M$$