Yes, the steps outlined in the hint are helpful for the model to reach the correct answer. Here's why:

1. Factorization of the Denominator (Step 1): The denominator $k^2 - 7k + 10$ factors as $(k - 5)(k - 2)$. This is a crucial first step in partial fraction decomposition, as it breaks the denominator into simpler factors.

2. Partial Fraction Decomposition (Step 2): The decomposition of $\frac{6}{(k - 5)(k - 2)}$ into $\frac{A}{k - 5} + \frac{B}{k - 2}$ is correctly performed here. The values for $A$ and $B$ are found using standard techniques (solving for coefficients), and it is clear that $A = 2$ and $B = -2$.

3. Writing the Sum (Step 3): The sum is then rewritten as a series in the form $\sum_{k=3}^{\infty} \left( \frac{2}{k - 5} - \frac{2}{k - 2} \right)$. This step is correct as it applies the partial fractions to form a series.

4. Telescoping Series (Step 4): The hint demonstrates the telescoping nature of the series. Expanding the terms shows that many terms will cancel out, which is a key property of such series.

5. Cancellations and Limits (Step 5): The cancellation is correctly done, leaving only the terms that do not cancel. The limit of the remaining terms as $n \to \infty$ tends to 0, which correctly accounts for the convergence of the series.

Conclusion:

The steps logically follow the process of solving a telescoping series, and the provided explanation is clear and correct. This would help the model reach the correct answer by accurately analyzing and simplifying the series.

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If you'd like, I can walk you through any step in more detail or clarify something further. Would you like to explore any specific part of the process?

Here are some related questions you might find interesting:

1. What are the general steps for solving a telescoping series?
2. How do we find the values of $A$ and $B$ in partial fraction decomposition?
3. Can you explain the method for finding the limit of a series at infinity?
4. What are some common mistakes when working with partial fractions?
5. How does the telescoping property simplify the calculation of infinite series?

Tip: When dealing with telescoping series, always carefully track which terms cancel out, as this is the key to simplifying the sum.