How long to catch up to a stream started 1 hour ago at 1.5x speed?

The easy way to do this doesn't require calculus, or even geometric series. Say it takes $t$ hours to catch up, so you have viewed $1+t$ hours of content at $\frac32$ speed. $$t=\frac23(1+t)\implies t=2$$

To make this easier to understand, I will solve this as if you are an object travelling at $1.5m/s$, and the stream is an object which started one hour before you travelling at $1m/s$ in the same direction. So you are travelling at $1.5\times$ the speed of the stream.

We know that disance ($d$), speed ($s$), and time ($t$), are related as follows: $$\Delta d=\Delta s\Delta t$$ You are trying to find time, so rearranging for $t$ gives $$\Delta t=\frac{\Delta d}{\Delta s}$$ The initial "distance" between you and the stream is $3600$ metres, based on a speed of $1m/s$ for one hour. So $$\Delta d=3600m$$ The difference between your speed and the stream's speed is $1.5m/s-1m/s=0.5m/s$. So $$\Delta s=0.5m/s$$ Now solving for $\Delta t$: $$\Delta t=\frac{3600m}{0.5m/s}=7200s$$ So, it will take you $7200s$, or $2$ hours, to catch up with the stream.

You are watching at rate $1.5t$. The stream is playing at a rate of $1 hr + 1t$. The intersection of these two lines occurs when $1.5t = 1 hr + 1t$, or $t = 2 hr$.

I don't recommend solving this as a geometric series, but it can be solved as one:

The first $60$ minutes is watched in $60/1.5 = 40$ minutes, the next $40$ minutes is watched in $60/1.5^2 = 40/1.5 = 26.66\ldots$ minutes, and so on. Writing this as a sum, we have

$$ T = {60\over1.5} + {60\over1.5^3} + {60\over1.5^3} + \cdots $$

This is often written using the summation symbol:

$$ \sum_{i=1}^\infty {60\over1.5^i} $$

There is a neat identity for infinite series like this,

$$ \sum_{i=\color{orange}0}^\infty ar^i = {a\over1-r} $$

Noting that this series starts with $0$, we add and subtract $60/1.5^0$ from out series:

$$ \color{blue}{-60/1.5^0} + \color{blue}{60/1.5^0} + \sum_{i=1}^\infty {60\over1.5^i} = \color{blue}{-60/1.5^0} + \sum_{i=\color{blue}0}^\infty {60\over1.5^i} = -\color{blue}{60} + \sum_{i=0}^\infty {60\over1.5^i} $$

With $a = 60$ and $r = 1/1.5$, we can use the identity to solve the equation

$$ -60 + \sum_{i=0}^\infty {60\over1.5^i} = -60 + {60\over 1 - 1/1.5} = -60 + 180 = 120 $$

Here is just a thought (NOT FULL ANSWER) :

Consider two particles A and B on the x-axis. A is at ($60,0$) while B is at origin. Both move towards the positive x-direction ; A with speed $1unit/min$ and B with $1.5unit/min$.

How much time will B take to catch up to A?

(HINT: Have you heard about relative motion?)