From wikipedia, the MGF of negative binomial is en.wikipedia.org/wiki/Negative_binomial_distribution#:~:text=In%20other%20words%2C%20the%20negative,failures%2C%20and%20successes%20are%20integers which is different from the MGF you have obtained using the geometric distribution. There are 2 different MGF for geometric distribution. en.wikipedia.org/wiki/Geometric_distribution. The one that fits the negative binomial distribution is the geometric distribution with the PMF of (1-p)^k * p.
At 18:38, the binomial coefficient is incorrect.
From wikipedia, the MGF of negative binomial is en.wikipedia.org/wiki/Negative_binomial_distribution#:~:text=In%20other%20words%2C%20the%20negative,failures%2C%20and%20successes%20are%20integers
which is different from the MGF you have obtained using the geometric distribution.
There are 2 different MGF for geometric distribution. en.wikipedia.org/wiki/Geometric_distribution.
The one that fits the negative binomial distribution is the geometric distribution with the PMF of (1-p)^k * p.