To determine whether the series

$\sum_{k=1}^{\infty} \frac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}}$

is convergent or divergent using the comparison test, we need to compare it with a known series that has a similar behavior as $k \to \infty$.

Step-by-Step Analysis

1. Identify Dominant Terms: For large $k$, the terms $k^3$ in the denominator and $\sqrt[3]{k}$ in the numerator dominate. So, the expression $\frac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}}$ behaves similarly to $\frac{\sqrt[3]{k}}{\sqrt{k^3}}$ as $k \to \infty$.

2. Simplify the Dominant Terms: $\frac{\sqrt[3]{k}}{\sqrt{k^3}} = \frac{k^{1/3}}{k^{3/2}} = k^{1/3 - 3/2} = k^{-7/6}.$

   This suggests that $\frac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}}$ behaves similarly to $k^{-7/6}$ for large $k$.

3. Comparison with a $p$-Series: The series $\sum \frac{1}{k^{7/6}}$ is a $p$-series with $p = \frac{7}{6}$. Since $p > 1$, this series converges.

4. Applying the Limit Comparison Test: To confirm, we can use the limit comparison test between $\frac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}}$ and $\frac{1}{k^{7/6}}$: $\lim_{k \to \infty} \frac{\frac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}}}{\frac{1}{k^{7/6}}} = \lim_{k \to \infty} \frac{k^{1/3}}{k^{3/2}} \cdot k^{7/6} = \lim_{k \to \infty} C = \text{constant} \neq 0.$

Since this limit yields a positive constant, we conclude by the limit comparison test that the given series converges.

Conclusion

The series

$\sum_{k=1}^{\infty} \frac{\sqrt[3]{k}}{\sqrt{k^3 + 4k + 3}}$

is convergent by comparison with the $p$-series $\sum \frac{1}{k^{7/6}}$.

Would you like a deeper breakdown of any step?

Here are five related questions to expand your understanding:

1. What is the difference between the direct comparison test and the limit comparison test?
2. How do you determine the dominant term in the numerator and denominator of a series term?
3. Why is the $p$-series test important for convergence and divergence analysis?
4. Can a series with a non-positive integer power in the denominator always be analyzed with a comparison test?
5. How does the convergence of $p$-series change as $p$ changes?

Tip: When using the comparison test, always simplify the dominant terms first to guide your choice of comparison series.