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Why the dynamic programming algorithm may not work in some cases?

Algorithm:

I think the working principle of the algorithm is a recursive or iterative algorithm. But can anyone tell me the scenario where the algorithm may not work? Like, when to use?

A:

Think about the general case. Let $a_i$ be the cost of the $i$-th step. Take $a_i = i^2$. Take $x_1 = 1, x_2 = 1/2, x_3 = 1/6, \dots$. Let $y_i = \dfrac{\sqrt{a_{i+1}} – \sqrt{a_i}}{\sqrt{a_i}}, i \ge 2$
Then $\sqrt{y_{i+1}} – \sqrt{y_i} = \dfrac{1}{\sqrt{a_{i+1}} + \sqrt{a_i}} \cdot \dfrac{\sqrt{a_{i+1}} – \sqrt{a_i}}{\sqrt{a_i}}, i \ge 2$.
The end is square-rooted.

A:

You don’t need to call the recursion in all cases. If your sequence has some properties, like being cyclic, monotonic, or satisfying certain recursion relations, then you may be able to prove that dynamic programming is unnecessary. For example, say you’re given a sequence of length $n$, with period $m$. Then for $k\in[0,n/m]$ you can make

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