To calculate the sum of the series given in the image, we need to analyze the series and then simplify it if possible.

The series is given by:

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

To find the sum, follow these steps:

1. Identify the general term: The general term is $\frac{\sin q}{q - 3} - \frac{\sin(q+2)}{q - 1}$.

2. Simplify the series: Let's denote the general term as $a_q = \frac{\sin q}{q - 3} - \frac{\sin(q+2)}{q - 1}$.

3. Write out the first few terms to look for a pattern:

   &\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 \end{align*}$$ Notice that the second term of each $$\frac{\sin(q+2)}{q-1}$$ has the potential to cancel out with part of the next term.

4. Telescoping nature: To see this more clearly, look at the expression: $\sum_{q=1}^{k} \left( \frac{\sin q}{q - 3} - \frac{\sin(q+2)}{q - 1} \right)$ Each $- \frac{\sin(q+2)}{q - 1}$ term will cancel out with $\frac{\sin(q+2)}{q+1 - 3}$ in the next term, creating a telescoping effect.

5. Evaluate the simplified series: After simplification, the remaining terms will be the first few and the last few terms that do not have counterparts to cancel with.

Let's write the terms that do not cancel:

• The initial terms $\frac{\sin 1}{1-3}$, $\frac{\sin 2}{2-3}$, and $\frac{\sin 3}{3-3}$.
• The final terms involving $\frac{\sin(k+1)}{k-2}$ and $\frac{\sin(k+2)}{k-1}$.

Therefore, the remaining terms are: $\sum_{q=1}^{3} \frac{\sin q}{q - 3} - \frac{\sin(1+2)}{1 - 1} + \frac{\sin(k+1)}{k - 2} - \frac{\sin(k+2)}{k - 1}$

Since the term $\frac{\sin(3)}{0}$ and $\frac{\sin(1+2)}{1-1}$ are undefined, this requires a more careful limit approach or consideration.

Would you like to proceed with the exact simplification or look into an alternative method?

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Would you like more details or have any questions? Here are some related questions that could expand the discussion:

1. What is a telescoping series and how does it help in simplifying sums?
2. Can all series be simplified using telescoping?
3. How can we handle terms in a series that result in undefined expressions?
4. What are some common techniques for evaluating trigonometric series?
5. How do we approach series where terms involve $\sin$ or $\cos$?
6. What is the importance of checking the initial terms in a series sum?
7. How can series be evaluated using limits?
8. What are some common pitfalls in simplifying complex series?

Tip: Always check for patterns in series terms; telescoping sums can significantly simplify the problem.