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The image contains a problem related to the limit of a series. Here is a transcription of the problem:

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**Exercise:**  
Consider the sum of the first \( n \) terms of the series \( S_n = \sum_{k=1}^{n} \frac{1}{k^p} \) with \( p > 1 \).

(a) Show that \( S_n \) is bounded for \( p > 1 \).

(b) Prove or disprove that \( S_n \) converges as \( n \to \infty \) for \( p = 1 \), \( p > 1 \), and \( p < 1 \).

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This problem involves series convergence and bounding. Let's break it down.

### (a) Showing that \( S_n \) is bounded for \( p > 1 \):
For \( p > 1 \), we need to show that the sum of the series is bounded as \( n \) increases. This involves finding an upper limit that the sum does not exceed.

#### Hint:
One approach is to compare \( S_n \) with an integral. For \( p > 1 \), \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) is a convergent p-series, which implies that the partial sums \( S_n \) are bounded.

### (b) Convergence analysis for different values of \( p \):
- **For \( p > 1 \)**: The series \( S_n \) converges as \( n \to \infty \).
- **For \( p = 1 \)**: The series \( S_n \) becomes the harmonic series, which diverges.
- **For \( p < 1 \)**: The series diverges more rapidly because the terms do not decrease fast enough.

If you need further explanation or help with solving the problem step-by-step, feel free to ask!

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Would you like detailed steps for proving part (a) or analyzing part (b)?

### Related Questions:
1. How can we use the integral test to determine if a series converges?
2. What is the difference between a convergent and divergent series?
3. How does the value of \( p \) affect the convergence of the p-series?
4. Can you give an example of a bounded but non-convergent sequence?
5. What is the significance of the harmonic series in mathematical analysis?

### Tip:
When dealing with series, always consider using convergence tests like the integral test, comparison test, and ratio test to analyze the behavior as \( n \) approaches infinity.

Consider the sum of the first n terms of the series S_n = Σ (1/k^p) for k=1 to n, with p > 1. (a) Show that S_n is bounded for p > 1. (b) Prove or disprove that S_n converges as n → ∞ for p = 1, p > 1, and p < 1.

Explore the convergence and boundedness of the series S_n = Σ (1/k^p) for different values of p. Understand the implications of the p-series test and learn to analyze series behavior for p < 1, p = 1, and p > 1. Suitable for university-level mathematics students.

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