How is ETS method equivalent to the MEAN method when $\alpha$ =0? I am learning time series analysis from Forecasting: Principles and Practice by Hyndman & Athanasopoulos. In chapter 8, the ETS (Error, Trend, Seasonality - a state space implementation of Exponential Smoothing) model is written as
$$\hat{y}_{T+1|T} = \alpha y_T + \alpha(1-\alpha) y_{T-1} + \alpha(1-\alpha)^2 y_{T-2}+ \cdots$$
where $\alpha$ is a smoothing parameter that takes the value between 0 and 1. When this $\alpha$ is 1 the ETS is equivalent to the NAIVE method and when this parameter takes on the value of 0, the ETS is equivalent to the MEAN method.
While I understand how ETS = NAIVE method when the value of $\alpha$ is 1, I fail to fully understand how this method becomes equivalent to the MEAN when $\alpha$ takes on the value of 0. There is explanation for why this is, in the given chapter but I find it a bit tricky to understand. I would appreciate it if someone could explain how come when $\alpha$=0, the ETS method is equivalent to the MEAN method.

You've missed the last term. The full expression is
$$\hat{y}_{T+1|T}  =  \sum_{j=0}^{T-1} \alpha(1-\alpha)^j y_{T-j} + (1-\alpha)^T \ell_{0},$$
so when $\alpha = 0$,
$$\hat{y}_{T+1|T}  = \ell_{0}.$$
If the parameter $\ell_0$ is estimated using least squares, we are minimizing
$$\sum_{t=1}^T (y_t - \ell_0)^2$$
which has minimum at
$$\hat{\ell}_0 = \frac{1}{T}\sum_{t=1}^T y_t.$$
