https://file.aiquickdraw.com/mathbot/file/172874691475486t4jyoo.jpg

The series in the image is:

\[
\sum_{k=1}^{\infty} \frac{100}{\sqrt{4k^2 + 10k}}
\]

To determine whether this series converges or diverges, we can apply the **Comparison Test** or the **Limit Comparison Test** by comparing it to a simpler known series. Let's simplify the behavior of the terms as \( k \) becomes large.

### Asymptotic Behavior for Large \( k \):
For large \( k \), the dominant term in the denominator \( \sqrt{4k^2 + 10k} \) will be \( \sqrt{4k^2} = 2k \). Thus, for large \( k \), the term inside the summation behaves approximately as:

\[
\frac{100}{\sqrt{4k^2 + 10k}} \approx \frac{100}{2k} = \frac{50}{k}
\]

This simplifies to a series that resembles:

\[
\sum_{k=1}^{\infty} \frac{50}{k}
\]

The series \( \sum_{k=1}^{\infty} \frac{1}{k} \) is the **harmonic series**, which is known to **diverge**.

### Limit Comparison Test:
We now perform the **Limit Comparison Test** by comparing the original series to \( \frac{1}{k} \).

Let:

\[
a_k = \frac{100}{\sqrt{4k^2 + 10k}} \quad \text{and} \quad b_k = \frac{1}{k}
\]

We calculate the limit:

\[
\lim_{k \to \infty} \frac{a_k}{b_k} = \lim_{k \to \infty} \frac{\frac{100}{\sqrt{4k^2 + 10k}}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{100k}{\sqrt{4k^2 + 10k}}
\]

For large \( k \), the denominator \( \sqrt{4k^2 + 10k} \) behaves like \( 2k \), so:

\[
\lim_{k \to \infty} \frac{100k}{2k} = 50
\]

Since the limit is a finite positive number (50), by the **Limit Comparison Test**, the given series behaves like the harmonic series \( \sum_{k=1}^{\infty} \frac{1}{k} \), which diverges.

### Conclusion:
The series \( \sum_{k=1}^{\infty} \frac{100}{\sqrt{4k^2 + 10k}} \) **diverges**.

---

Would you like more details on the comparison test or another topic?

Here are 5 related questions you might find useful:
1. How does the comparison test work in more detail?
2. What are examples of convergent series?
3. Can the root test be used to check series convergence?
4. How do other divergence tests (e.g., ratio test) apply to this problem?
5. How do we prove the harmonic series diverges?

**Tip:** Always compare series to known simple series to determine their behavior for large values of \( k \).

Determine if the series ∑ (from k=1 to ∞) 100 / √(4k^2 + 10k) is convergent or divergent.

Learn how to determine the convergence or divergence of the series ∑ (from k=1 to ∞) 100 / √(4k^2 + 10k) using the Limit Comparison Test. Explore its asymptotic behavior and comparison with the harmonic series, which leads to the conclusion that the given series diverges. Suitable for college-level calculus students.

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