You're correct that terms involving undefined expressions like $\frac{\sin 3}{3-3}$ need careful handling. Let's take a detailed step-by-step approach to clarify this.

Given series: $\sum_{q=1}^{k} \left( \frac{\sin q}{q - 3} - \frac{\sin(q+2)}{q - 1} \right)$

To simplify, let's consider writing out the terms to observe the potential telescoping effect and handle the problematic terms explicitly.

1. Write out the terms:

&\left( \frac{\sin 1}{1 - 3} - \frac{\sin(1+2)}{1 - 1} \right) + \left( \frac{\sin 2}{2 - 3} - \frac{\sin(2+2)}{2 - 1} \right) + \left( \frac{\sin 3}{3 - 3} - \frac{\sin(3+2)}{3 - 1} \right) + \cdots \\ &+\left( \frac{\sin k}{k - 3} - \frac{\sin(k+2)}{k - 1} \right) \end{align*}$$ 2. **Simplify and identify undefined terms**: - The term $$\frac{\sin 1}{1 - 3}$$ is defined as $$\frac{\sin 1}{-2}$$. - The term $$\frac{\sin 2}{2 - 3}$$ is defined as $$\frac{\sin 2}{-1}$$. - The term $$\frac{\sin 3}{3 - 3}$$ is undefined ($$\frac{\sin 3}{0}$$) and must be carefully treated. - The term $$\frac{\sin (1+2)}{1 - 1}$$ is undefined ($$\frac{\sin 3}{0}$$) and must be carefully treated. To handle these undefined terms, we'll exclude them from direct evaluation and focus on the valid terms first. 3. **Focus on valid terms**: $$\left( \frac{\sin 1}{1 - 3} \right) + \left( \frac{\sin 2}{2 - 3} \right) + \left( \frac{\sin 4}{2 - 1} \right) + \left( \frac{\sin 5}{3 - 1} \right) + \cdots + \left( \frac{\sin k}{k - 3} \right) + \left( \frac{\sin (k+1)}{k - 2} \right) - \left( \frac{\sin (k+2)}{k - 1} \right)$$ 4. **Cancel out terms that simplify**: $$\left( \frac{\sin q}{q - 3} - \frac{\sin (q+2)}{q - 1} \right)$$ Notice the pattern that forms the telescoping series: $$\frac{\sin 4}{1} + \frac{\sin 5}{2} + \cdots + \frac{\sin k}{k - 3} + \left( \frac{\sin (k+1)}{k - 2} - \frac{\sin (k+2)}{k - 1} \right)$$ 5. **Evaluate and combine remaining terms**: Remaining terms: $$-\frac{\sin 1}{2} - \sin 2 + \frac{\sin (k+1)}{k - 2} - \frac{\sin (k+2)}{k - 1}$$ Thus, the valid expression for the sum would be: $$-\frac{\sin 1}{2} - \sin 2 + \frac{\sin (k+1)}{k - 2} - \frac{\sin (k+2)}{k - 1}$$ Would you like to evaluate this further for specific $$k$$ values or need further clarification? --- Would you like more details or have any questions? Here are some related questions that could expand the discussion: 1. How do you handle undefined terms in a series? 2. What is the significance of the telescoping effect in simplifying series? 3. How can you verify the convergence of a series? 4. Are there other methods to simplify complex series involving trigonometric functions? 5. What role does the initial term play in the sum of a series? 6. How can limits be used to evaluate terms that are undefined? 7. What are common mistakes to avoid when dealing with infinite series? 8. How can you apply these techniques to other types of series? **Tip:** Always break down the series and check for potential simplification patterns like telescoping to make the process more manageable.