Trains purposely crashing into each other to kill a bee

You might infer that the problem is the one I’m going to explain soon.

This puzzle has a lot of versions, but for simplicity’s sake, I’ll be using AskAMathematician‘s version.

Train A and Train B are 700 miles apart (not kilometres :v) and the two trains are facing each other (flashbacks of Fast & Furious 7), Train A at 85 mph, and Train B at 55 mph. A bee (let’s call it AskMathematician) at the nose of Train A, can go at the speed of, not light, but 110 mph. When AskMathematician hits Train B, it turns 180 degrees and flies towards Train A, and vice versa. When the two trains hit each hit other and AskMathematician dies:

Before AskMathematician gets absolutely squashed, how much distance would AskMathematician have covered, in miles?

Now, the “mathematician” way to solve this is by actually summing up the distance covered, but I’ll cover it later on. The “swag” method is:

85 + 55 = 140, 700/140 = 5 (hours), 5 * 110 = 550 miles. :v

EDIT: I messed up the name. It’s AskAMathematician, not AskMathematician. :v ARE YOU KIDDING ME?!?!

The mathematician was would calculate the time to get from Train A’s starting point to the collision point , then do the same, over and over until the trains collide with each other.

Let’s add a happy ending. If AskMathematician sees the future and flies up one mile before the trains collide (headlines: Two trains collide into each other, AskMathematician barely escaped), how long will that take and how far? Five hours, thirty-two seconds, 727 milliseconds, approximately, and 551 miles. :v

I might go into further detail on the “infinite sum” solution tomorrow.

Also, I worked on this yesterday, but couldn’t finish it, so here we are.