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We are asked to determine whether the given series converges and, if so, to find its value. The series in question is:

\[
\sum_{k=1}^{\infty} \frac{k(k + 14)}{(k + 3)^2}
\]

### Step 1: Analyze the behavior of the terms for large \( k \)

To determine if the series converges, we first examine the behavior of the general term for large \( k \). When \( k \) is large, the dominant term in both the numerator and denominator will contribute to the behavior of the series.

The general term is:

\[
\frac{k(k + 14)}{(k + 3)^2}
\]

For large \( k \), this behaves approximately like:

\[
\frac{k(k)}{(k)^2} = \frac{k^2}{k^2} = 1
\]

Thus, the series behaves similarly to the harmonic series \( \sum \frac{1}{k} \), which is divergent. This suggests that the given series may also diverge.

### Step 2: Use the Limit Comparison Test

To confirm, we can use the **Limit Comparison Test** with a simpler series that we know diverges. Let's compare the given series with the harmonic series \( \sum \frac{1}{k} \).

We compute the limit of the ratio of the terms of the given series and the harmonic series as \( k \to \infty \):

\[
\lim_{k \to \infty} \frac{\frac{k(k + 14)}{(k + 3)^2}}{\frac{1}{k}} = \lim_{k \to \infty} \frac{k^2 + 14k}{k(k + 3)^2}
\]

Simplifying:

\[
= \lim_{k \to \infty} \frac{k(k + 14)}{(k + 3)^2} = \lim_{k \to \infty} \frac{k^2 + 14k}{k^2 + 6k + 9}
\]

Dividing both the numerator and the denominator by \( k^2 \):

\[
= \lim_{k \to \infty} \frac{1 + \frac{14}{k}}{1 + \frac{6}{k} + \frac{9}{k^2}} = 1
\]

Since the limit is a positive finite number, the given series behaves like \( \sum \frac{1}{k} \), which diverges. Therefore, the original series **diverges**.

### Conclusion

The series:

\[
\sum_{k=1}^{\infty} \frac{k(k + 14)}{(k + 3)^2}
\]

**diverges**.

---

Would you like to explore more details, or have any further questions on this problem?

### Related Questions:
1. What is the Limit Comparison Test and how is it applied?
2. What is the harmonic series, and why does it diverge?
3. Can you explain the behavior of a series in terms of dominant terms for large \( k \)?
4. How does the ratio test compare to the limit comparison test?
5. Are there any techniques for transforming a series to make its convergence more apparent?

**Tip:** When analyzing series, it's often useful to approximate terms for large values of \( k \) to get an idea of their behavior.

Consider the series ∑ (k(k + 14))/(k + 3)² from k = 1 to infinity. Determine whether the series converges, and if it converges, determine its value.

Explore the convergence of the infinite series k(k + 14) / (k + 3)² using the Limit Comparison Test. Learn how the behavior of the series resembles the harmonic series, leading to divergence. Detailed explanation for undergraduate math students.

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