function*                                    $\alpha_{(k)} = \sum_{j=1}^k\alpha_j \prod_{i=1}^{j-1}\left(1-\alpha_i\right)$
              *overall power*                                 $1-\beta(\theta) = \sum_{k=1}^K [1-\beta_k(\theta)] \prod_{j=1}^{k-1}\left[\beta_j(\theta)\right]$
           *stage-specific power*           $  1-\beta_k(\theta) = \text{Pr}_{\theta}\left( Z_{(k)} > c_k \big| \cap_{j=1}^{k-1} \left\{ Z_{(j)}\le c_{j} \right\}, \theta \right)$
             *cumulative power*                       $1-\beta_{(k)}\left(\theta\right) = \sum_{j=1}^k[1-\beta_j(\theta)] \prod_{i=1}^{j-1}\left[\beta_i(\theta)\right]$
           *overall type 2 error*                                                                       $\beta(\theta)$
       *stage-specific type 2 error*                                                                   $\beta_k(\theta)$
        *$\beta$-spending function*                                                            $\beta_{(k)}\left(\theta\right)$
     *stage-specific rejection region*              $ {\cal{R}}_k = \left\{Z_{(k)} > c_k \right\} \cap \left\{\cap_{j=1}^{k-1} \left\{ Z_{(j)}\le c_{j} \right\} \right\}$
   *stage-specific rejection probability*                                        $  \text{Pr}_{\theta}\left({\cal{R}}_k \vert \theta \right)$

The use of both $\alpha-$ and $\beta-$ spending functions was suggested by @pampallona2001 for use with a single alternative hypothesis at predetermined samples sizes. But the use of a sequence of ordered alternative hypotheses with the same power to determine sample sizes is new. In this optimization problem, there are $K+1$ (marginal) restrictions (type 1 error $=\alpha$ under $H_0$ and power $=1-\beta$ under each alternative) and $2K$unknown quantities $\{n_1,\ldots,n_K,c_1,\ldots,c_K\}$. The $Z_{(k)}$ pivots do not have to be normal random variables. More details are given through the example that follows.

One Simple Option
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It is common to select an $\alpha$-spending function that determines conditional type $1$ errors while keeping the overall type $1$ error equal to $\alpha$. For simplicity of exposition, for all $k$, assume that $\alpha_k 