One way to effectively engage prospects is to use the Talk-Pivot-Close approach. The idea here is simple: you must earn the right to ask someone for their time and attention on LinkedIn.
You do that by providing some free tips, advice, content, etc., aimed at solving a pain point your prospect experiences in his or her business. Perhaps start the relationship off with a casual, one-on-one conversation via LinkedIn; you can share your free tips or resources as a natural extension of the conversation.
Offer something of value right away without asking for anything in return (a phone call, a meeting, etc.). By building the relationship with the prospect in this way, you can make a bigger “ask” later on.