I know rapid rise in PSA is a concern but this is a huge rise in short time...
(finishing your sentence for you)
"...which is usually indicative of a prostate infection."
A lot--too may--urologists jump to a prostate biopsy without trying first to treat a prostate biopsy...which is the AUA "best practice." Way too many unnecessary biopsies, which are not, themselves, without risk. You might want to ask why he is not following AUA "best practices." Google: AUA PSA best practices
So being over 50, you've probably got at least some small amount of prostate cancer growing. The percentage of men having it (mostly indolent) is roughly equal to their age. 70% of 70-year olds, etc.
So a prostate biopsy for someone like you can be somewhat of a fishing expedition. Be careful of the possible "slippery slope" you will be tempted to get on. Over-treatment for favorable-risk PC remains rampant in the US. (see the recent discussions here on "regret")
Never too early to start learning more about
your prostate, and the somewhat inevitable coming (with age) of prostate cancer. I strongly recommend you begin by reading the sticky thread (top of page) titled "Newly diagnosed with PC? - read this thread first."