https://file.aiquickdraw.com/mathbot/file/172576533172387w3kg0x.jpg

Let's focus on part (b) of the problem from your image. It seems like the task is to write a pseudocode for testing whether the harmonic series diverges, given the harmonic series:

\[
H_n = \sum_{k=1}^{n} \frac{1}{k}
\]

The harmonic series is divergent, meaning as \( n \) increases, the sum does not settle to a finite value. The pseudocode needs to compute this sum and check how it grows.

Here's a possible pseudocode outline for testing divergence by checking how the sum behaves as \( n \) becomes large:

### Pseudocode to Check Divergence of Harmonic Series

```
Set n = 1
Set Sum = 0

Repeat:
    Add 1/n to Sum
    Print current value of Sum and n
    Increase n by 1
    If n is very large (e.g., 1000000 or more), stop
```

### Explanation:
1. Start with \( n = 1 \) and an initial sum \( \text{Sum} = 0 \).
2. In each iteration, add \( \frac{1}{n} \) to the sum.
3. Keep printing the sum and the current value of \( n \).
4. The code will run until a large value of \( n \) is reached (a stopping criterion should be introduced based on how large \( n \) should be).

### Python Example:

```python
n = 1
Sum = 0

while n <= 1000000:
    Sum += 1 / n
    print(f"n = {n}, Harmonic sum = {Sum}")
    n += 1
```

This will compute the harmonic series up to \( n = 1,000,000 \), and you will observe that the sum keeps increasing, confirming that the series diverges.

Do you have any questions or need more detailed explanation on this part?

---

**Relative Questions:**
1. What is the mathematical definition of a divergent series?
2. How fast does the harmonic series grow compared to other divergent series?
3. What is Euler’s constant, and how is it related to the harmonic series?
4. How can you modify the pseudocode to calculate a partial sum more efficiently?
5. What is the difference between conditional and absolute convergence?

**Tip:** When writing a pseudocode for a mathematical algorithm, ensure that you clearly define stopping conditions to prevent infinite loops.

Learn how to test the divergence of the harmonic series using pseudocode. This problem demonstrates how to calculate the partial sums of the harmonic series and check its behavior. Suitable for undergraduate-level mathematics or algorithm students.

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