TITLE: Is it acceptable to say that a divergent series that tends to infinity is 'equal to' infinity?
QUESTION [22 upvotes]: 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?

REPLY [6 votes]: A plea from a high-school teacher:  Do not use $=\infty$ anywhere in any school course.  The idea of infinity already causes a great deal of confusion amongst schoolchildren (as well as fascination), and our best approach is to say firmly that, 'Infinity is not a number, but an idea, and the notation $\infty$ is used as shorthand in a few standard pieces of notation.'
Yes, I know that everything in mathematics is an idea, and that every mark on the page is notation, but leave that to university philosophy classes.  What we don't want is monstrosities such as $\infty - \infty = 0$ and $\infty / \infty = 1$, which school-children routinely arrive at if we give even a hint that infinity is a number.
(I wouldn't be using $=\infty$ in undergraduate courses either.  Without a rigorous axiomatic understanding of mathematics, plus some projective geometry, it is misunderstood and misused.)