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To solve question 34, we need to evaluate the limit of the sum:

\[
\lim_{n \to \infty} \sum_{k=1}^{n} \left( \sin \frac{\pi}{2k} - \cos \frac{\pi}{2k} - \sin \frac{\pi}{2(k + 2)} + \cos \frac{\pi}{2(k + 2)} \right)
\]

Let's analyze and simplify the terms in the summation:

1. The terms \(\sin \frac{\pi}{2k}\) and \(\sin \frac{\pi}{2(k+2)}\) are close to each other for large \(k\).
2. Similarly, \(\cos \frac{\pi}{2k}\) and \(\cos \frac{\pi}{2(k+2)}\) are close for large \(k\).

The key idea is to recognize that the sum has a telescoping nature, where many terms will cancel out as \(n \to \infty\).

### Simplification

Notice that:

\[
\sum_{k=1}^{n} \left( \sin \frac{\pi}{2k} - \sin \frac{\pi}{2(k + 2)} \right)
\]

and 

\[
\sum_{k=1}^{n} \left( \cos \frac{\pi}{2(k + 2)} - \cos \frac{\pi}{2k} \right)
\]

are similar to each other, with opposite signs, and will cancel most terms as \(n \to \infty\). The first few terms and the last few terms are the only ones that do not cancel out. 

Specifically, the first term for \(\sin \frac{\pi}{2k}\) and \(\cos \frac{\pi}{2(k+2)}\) will remain. Thus, the limit boils down to the initial terms that do not get canceled.

### Conclusion

Evaluating the residual terms, we get:

\[
\lim_{n \to \infty} \left( \sin \frac{\pi}{2} + \cos \frac{\pi}{2} \right) = 1
\]

Hence, the answer is \(1\).

If you have any further questions or need more details, feel free to ask!

### Related Questions

1. What is the general strategy for handling limits involving trigonometric functions?
2. How do you recognize a telescoping series?
3. What are the properties of sine and cosine functions that simplify in limits?
4. How can we approximate trigonometric functions for small angles?
5. What are common techniques to evaluate limits involving summations?
6. What is the significance of the terms that do not cancel in a telescoping series?
7. How do trigonometric identities help in solving limits?
8. What are some common pitfalls in evaluating limits involving infinite series?

**Tip:** For problems involving telescoping series, carefully identify the terms that do not cancel out, as they are crucial for finding the limit.

Explore how to solve question 34 involving trigonometric limits and telescoping series. This problem requires understanding of limits, trigonometric identities, and series summation properties. Suitable for advanced undergraduate students studying calculus and mathematical analysis.

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