1. Let's analyze the given problem step-by-step to clarify the notation and expressions. 2. The series given is $$\sum_{n=0}^{\infty} (-1)^n c_n \leq 1$$. This means the alternating series formed by the terms $c_n$ satisfies an upper bound of 1. 3. The limits $$\lim_{n \to +\infty} a_n \geq b_n \geq \ldots$$ mean there is a sequence $a_n$ that is bounded below by $b_n$, and so on, indicating a non-increasing chain of inequalities for sequences as $n$ approaches infinity. 4. For the product terms, $$a_n b_n n (n-1)$$ seems like a general term possibly related to the series or integral. 5. The integral expression $$\lim_{n \to +\infty} \int a_n \left( \int x^n dx \right)$$ involves nested integrals with sequence terms and powers of $x$. 6. To interpret and solve concretely: - The inner integral $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$. - Therefore, $$\int a_n \left( \int x^n dx \right) = a_n \left( \frac{x^{n+1}}{n+1} + C \right)$$. - Taking limit $$n \to +\infty$$ depends on the behavior of $a_n$ and the variable $x$. 7. If $a_n$ converges to $a$, and $|x| < 1$, then $$\lim_{n \to +\infty} a_n \frac{x^{n+1}}{n+1} = 0$$ since $x^{n+1}$ goes to 0 faster than $n+1$ grows. 8. Without explicit forms of $a_n$, $b_n$, and $c_n$, we can't evaluate the exact sum or limit. 9. But structurally, the problem examines an alternating series with bounded sum, sequences with inequalities, and nested integrals involving powers of $x$ and these sequences. 10. If you provide explicit forms or clearer expressions, we can find closed-form solutions or convergence behavior. Final interpretation is qualitative due to limited information.