If you're watching a live stream and n number of minutes behind the live feed, how much time would it take while watching at s speed to catch up? Sometimes I'll tune into YouTube livestreams a few minutes late, but I want to watch the whole thing and catch up with the live feed, so I'll start from the beginning at 2x speed or 1x speed. I'd like to find a way to calculate exactly how much time it will take for me to reach the live feed.
So if I start watching 20 minutes behind at 2x speed, after 20 minutes of watching, I'll be 10 minutes behind, after 10 minutes, I'll be 5, etc.
I would guess this would look something like this
$ total = \frac{20}{2} + \frac{10}{2} + \frac{5}{2} + ... $
How would you create a general equation for this? Could you use summation, product, or something like a Taylor series? I recall an old VSauce Video about SuperTasks which feels relevant here as well.

Say you are $t$ minutes behind and you start to watch the stream at $m$x speed.So after $\frac{t}{m}$ time you will be $\frac{t}{m}$ behind ,after $\frac{t}{m^2}$ time you will be $\frac{t}{m^2}$ behind and so on
So total time$=\frac{t}{m}+\frac{t}{m^2}+\frac{t}{m^3}+.......=\frac{\frac{t}{m}}{1-\frac{1}{m}}=\frac{t}{m-1}$
Anecdote
This is a really interesting way to visualise the Zeno's Dichotomy paradox which you can find here
