“Is there any way I can convince you to reconsider?”
You are a salesperson and one of your clients has told you that she's canceling her account and moving over to a competitor's product. You ask her this because you don't want to lose the business.
Is there any way I can convince you to reconsider?
"Convince" means to make someone decide something based on what you say to them. You can use "convince" without "to". For example, the following can be said if it's already clear from the conversation what you have been convinced of:
He convinced me.
Use "to (do something)" to tell an action that the person was convinced to do. Use "that (clause)" to express that the person was convinced of an idea:
Ed convinced me that the fastest way to get to New York was Highway 95.
"Reconsider" means to think about your decision again, and possibly change it. In the example sentence above, "reconsider" is used as an intransitive verb. But it's usually followed by a noun:
Please reconsider my offer.
In the example above, both the speaker and the listener understand what is being reconsidered, so it's left out of the sentence:
Is there any way I can convince you to reconsider (cancelling your account)?