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

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 $p < 1$.

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!

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.