Is it possible to upgrade Apache Cassandra 2.1.9+ to Apache Cassandra 3.1+ directly?
The release notes for 3.0 mention direct upgrades need a minimum of Apache Cassandra 2.1.9+, but all further releases of Apache Cassandra don't mention whether an intermediate version is needed.
Yes, you can upgrade from Cassandra 2.1.9 (or higher) to Cassandra 3.1 (or higher).
As stated in the DataStax dev blog in June of 2015, Cassandra moved to a "tick-tock" release cycle with version 3. The details of which you can get from the link, but the main point is that the release structure of 3.x is not the same as it was with 2.x.
Cassandra 2.0, 2.1, and 2.2 were different enough in terms of features and storage modifications that they required a structured upgrade path. Cassandra 3.x does not. You can upgrade directly to 3.1, 3.4, or 3.10 (just released). As long as you meet the 3.0 upgrade requirements (Cassandra version >= 2.1.9) you don't need to worry about intermediate upgrades with the 3.x line.