Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.
1. **Restate the problem and given conditions:**
   We are given \( M, \alpha, \beta \in \mathbb{R} \) with \( M > 0 \) and \( \alpha, \beta \in (0,1) \). We need to determine the smallest real \( R > 1 \) for which there exists an \( R \)-*inoceronte* sequence \( \{ C_n \}_{n \geq 0} \) such that \( \sum_{n=1}^{\infty} \beta^n C_n^\alpha \) diverges.

2. **Define \( R \)-*inoceronte* sequence:**
   A sequence \( \{ C_n \}_{n \geq 0} \) is \( R \)-*inoceronte* if:
   \[
   \sum_{i=1}^n R^{n-i} C_i \leq R^n \cdot M \quad \text{for all } n \geq 1.
   \]
   This can be rewritten as:
   \[
   \sum_{i=1}^n R^{-i} C_i \leq M.
   \]

3. **Claim 1: \( M \) is irrelevant:**
   If a sequence \( \{ C_i \} \) satisfies the condition for some \( M > 0 \), then the sequence \( \left( \frac{N}{M} C_i \right) \) satisfies the condition for \( M = N \). Therefore, we only need to consider whether \( \sum_{n \geq 1} R^{-n} C_n \) converges or not.

4. **Claim 2: If \( R = \beta^{-\frac{1}{\alpha}} \), such a sequence exists:**
   - Let \( C_n R^{-n} = \frac{1}{n^{\frac{1}{\alpha}}} \). Then:
     \[
     C_n = \frac{R^n}{n^{\frac{1}{\alpha}}} = \sqrt[\alpha]{\frac{1}{n \beta^n}}.
     \]
   - Check condition (i):
     \[
     \sum_{n \geq 1} \frac{1}{n^{\frac{1}{\alpha}}} < \infty,
     \]
     which is true since \( \alpha \in (0,1) \).
   - Check condition (ii):
     \[
     \sum_{n \geq 1} \beta^n C_n^\alpha = \sum_{n \geq 1} \frac{1}{n},
     \]
     which diverges.

5. **Claim 3: If \( R < \beta^{-\frac{1}{\alpha}} \), such a sequence does not exist:**
   - Let \( R = \lambda \beta^{-\frac{1}{\alpha}} \) with \( 0 < \lambda < 1 \).
   - Write \( C_n = R^n d_n \) for some sequence \( \{ d_n \} \).
   - Condition (i) implies:
     \[
     \sum_{n \geq 1} R^{-n} C_n < \infty \implies \sum_{n \geq 1} d_n < \infty.
     \]
   - For large \( n \), \( d_n < 1 \). Condition (ii) must hold:
     \[
     \sum_{n \geq 1} \beta^n \cdot \lambda^{\alpha n} \beta^{-n} d_n^\alpha = \infty \implies \sum_{n \geq 1} (d_n \lambda^n)^\alpha = \infty.
     \]
   - Since \( d_n \lambda^n < \lambda^n \) for \( n > N \):
     \[
     \underbrace{\sum_{1 \leq n \leq N} (d_n \lambda^n)^\alpha}_{C} + \sum_{N < n} \lambda^{\alpha n} = \infty,
     \]
     which is a contradiction since \( \lambda \in (0,1) \).

6. **Conclusion:**
   Claims 2 and 3 together imply that the smallest \( R \) for which such a sequence exists is \( R = \beta^{-\frac{1}{\alpha}} \).

The final answer is \( \boxed{ R = \beta^{-\frac{1}{\alpha}} } \).