### The Cake Eating Problem in Sage

Im interested in running a basic cake eating problem in sage. In its recursive formulation we have to solve the following bellman equations:
$$v(k_t)=ln(k_t-k_{t+1})+\beta v(k_{t+1}), \ \ \ \beta\in(0,1)$$
The algorithm for solving this problem is as follows:

Step 1: Take an initial guess of $v(k_{t+1})=0$

Step 2: Solve for the maximum of $v(k_t)$ (in the first iteration this is ~~$v(k_t)=ln(k_t-k_{t+1})$)~~$v(k_t)=ln(k_t-k_{t+1})$, the maximum here is simply where $k_{t+1}=0$ thus $v(k)=ln(k_{t})$ )

Step 3: Using our maximum for $v(k_t)$ iterate it forward and update our bellman (in this case we have $v(k_t)=ln(k_t-k_{t+1})+\beta ~~[ln(k_{t+1}-k_{t+2})]$)~~[ln(k_{t+1}]$)

Step 4: Maximize this updated equation and repeat until convergence.

Any help is appreciated.