The claim sounds ridiculous to say the least.
There are however cases of alternating series that have different answers (not really, just that they are claimed to), but the key issue to realize about those is that you can't just re-arrange terms and expect to get a proper answer when you have an infinite series.
If a series converges then it has only one answer. If it diverges, then it does not have one specific answer and the final answer does not exist as a fixed value.
If you can point out how he proves this then we can take a closer look at it and find where he is tripping up.