We are asked to use the Direct Comparison Test to determine whether the given series

$\sum_{n=1}^{\infty} a_n = \sum_{n=1}^{\infty} \frac{9n^7 + 7 \tan^{-1}(n)}{3n^8 - 2}$

converges or diverges. To use the Direct Comparison Test, we need to find an appropriate comparison series $\sum_{n=1}^{\infty} b_n$ of the form

$\sum_{n=1}^{\infty} b_n = \sum_{n=1}^{\infty} c \cdot \frac{1}{n^p},$

and identify constants $c$ and $p$. Let’s work through it step-by-step.

---

Step 1: Simplify the Given Series as $n \to \infty$

We are given:

$a_n = \frac{9n^7 + 7 \tan^{-1}(n)}{3n^8 - 2}.$

For large $n$, the terms $9n^7$ and $3n^8$ dominate in the numerator and denominator respectively. So, the behavior of the sequence $a_n$ can be approximated by:

$a_n \approx \frac{9n^7}{3n^8} = \frac{9}{3} \cdot \frac{1}{n} = 3 \cdot \frac{1}{n}.$

This suggests that our comparison series might look like:

$b_n = 3 \cdot \frac{1}{n^p}.$

---

Step 2: Identify the Comparison Series

Thus, the comparison series has:

$b_n = 3 \cdot \frac{1}{n^1} = \frac{3}{n}.$

This is a p-series with $p = 1$.

---

Step 3: Apply the Direct Comparison Test

• p-series test: The series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges because $p = 1$.

• Now, we check if $a_n \leq b_n$ for all $n \geq 1$. For large $n$, we already saw that:

  $a_n \approx \frac{3}{n}.$

  So, for sufficiently large $n$, we have $a_n \leq b_n$.

---

Step 4: Conclusion

Since $\sum_{n=1}^{\infty} b_n = \sum_{n=1}^{\infty} \frac{3}{n}$ diverges, and $a_n \leq b_n$ for all $n \geq 1$, the given series $\sum_{n=1}^{\infty} a_n$ also diverges by the Direct Comparison Test.

---

Final Answer

• $c = 3$
• $p = 1$
• $\sum_{n=1}^{\infty} b_n$ diverges.
• Therefore, $\sum_{n=1}^{\infty} a_n$ also diverges by the Direct Comparison Test.

---

Do you want further details or have any questions?

Here are five related questions you may find useful:

1. How does the p-series test determine convergence?
2. What are the conditions for using the Direct Comparison Test?
3. Can the Limit Comparison Test be applied here instead?
4. What happens when $p > 1$ for p-series?
5. How does $\tan^{-1}(n)$ behave asymptotically for large $n$?

Tip: When comparing series, always focus on the dominant terms as $n \to \infty$.