The log-likelihood function for the Geometric distribution for the sample {x1, …, xn} is
The MLE value is achieved when
which is the same value as from the method of moments (see Method of Moments).
References
Forbes, C., Evans, M., Hastings, N., Peacock, B. (2011) Statistical distribution. Wiley
https://www.academia.edu/49056503/Statistical_distributions
Siegrist, K. (2022) Maximum-Likelihood
https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_(Siegrist)/07%3A_Point_Estimation/7.03%3A_Maximum_Likelihood
Millard, S. P. (2023) Estimate probability parameter of a geometric distribution
https://search.r-project.org/CRAN/refmans/EnvStats/html/egeom.html