There are widespread fears that conversational AI could soon exert unprecedented influence over human beliefs. Here, in three large-scale experiments (N=76,977), we deployed 19 LLMs-including some post-trained explicitly for persuasion-to evaluate their persuasiveness on 707 political issues. We then checked the factual accuracy of 466,769 resulting LLM claims. Contrary to popular concerns, we show that the persuasive power of current and near-future AI is likely to stem more from post-training and prompting methods-which boosted persuasiveness by as much as 51% and 27% respectively-than from personalization or increasing model scale. We further show that these methods increased persuasion by exploiting LLMs' unique ability to rapidly access and strategically deploy information and that, strikingly, where they increased AI persuasiveness they also systematically decreased factual accuracy.