I give this answer as mathematician: The first (as far as I know) documented instance of the symbol being used as a representative for the concept of infinity can be found in the book linked below.

De sectionibus conicis nova methodo expositis tractatus. John Wallis, 1655 (Bayrische Staatsbibliothek/ Bavarian state library)

Look on page 4. Wallis defines the "∞" symbol as concept of infinity (actually he defines ∞ as an "infinitely large number"). A few other mathematicians used other symbols but the concept stuck.

A subsequent question I have is, where the origin of the ∞ lies. I do not think Wallis decision to use this symbol for this purpose was arbitrary - or that others also starting to use ∞ for "infinite" was arbitrary.

So: "What is the symbolism and origin of the symbol ∞ before it was defined as representative for the concept of infinity [in 1655 by John Wallis]?"

€ replaced the googlebooks link with one of the Bavarian state library (where the physical body of this scan is.)