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The problem given is in the context of operator theory, specifically dealing with bounded operators on a Hilbert space \( H \), denoted \( B(H) \). The series to prove convergence for is:

\[
\sum_{n=0}^{\infty} A^n C B^n
\]
where \( A \) and \( B \) are bounded operators in \( B(H) \), and \( C \) is another operator.

### Approach to proving convergence:
To show that this series converges, we typically check conditions under which the operators \( A \) and \( B \) satisfy norms that decay sufficiently for large \( n \). Some steps we can consider include:

1. **Boundedness of Operators**: Since \( A, B \in B(H) \), they are bounded operators. Hence, we can denote their operator norms by \( \|A\| \) and \( \|B\| \), respectively.

2. **Convergence Criteria**: A common method for showing convergence of a series involving operators is to check if the norms \( \|A^n C B^n\| \) tend to zero as \( n \to \infty \), or if they are summable. A sufficient condition for convergence could be if \( \|A\| \cdot \|B\| < 1 \), which would ensure that \( A^n \) and \( B^n \) decay exponentially.

3. **Gelfand’s Theorem (in Operator Theory)**: If \( A \) and \( B \) are contractions, meaning \( \|A\|, \|B\| < 1 \), the powers of these operators \( A^n \) and \( B^n \) will tend to zero as \( n \to \infty \), ensuring that the series converges.

4. **Neumann Series Analogy**: If the operators \( A \) and \( B \) are contractions, the series behaves similarly to a Neumann series, which converges when the norm of the involved operator is less than 1.

To provide a detailed proof, one would need to verify these assumptions about the norms of the operators and apply results from operator theory.

Would you like further details on how to rigorously apply these ideas to complete the proof? Here are some follow-up questions that may help extend the understanding:

1. What is the significance of bounded operators in the convergence of infinite series?
2. How do operator norms affect the behavior of series like this one?
3. Could we generalize the result if \( A \) and \( B \) are not contractions?
4. How would this series behave if \( A \) and \( B \) commute?
5. What alternative techniques could we use to test the convergence of such operator series?

**Tip:** In operator theory, checking the norm condition \( \|A\| \cdot \|B\| < 1 \) can greatly simplify convergence proofs for series involving powers of operators.

Soient A et B deux opérateurs dans B(H). Montrer que la série Σ₀ⁿ Aⁿ C Bⁿ est convergente.

Explore how to prove convergence of a series Σ₀ⁿ Aⁿ C Bⁿ with bounded operators A and B in the space B(H) of Hilbert space operators. The solution involves assessing operator norms, applying Gelfand's Theorem, and understanding Neumann series analogy. Suitable for advanced university mathematics courses.

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