To solve this problem using the Root Test to determine whether the series $\sum_{n=1}^{\infty} \left( \frac{8n^3 - n - 7}{2n^2 + n + 4} \right)^n$ converges or diverges, we proceed as follows:

The Root Test states:

• For a series $\sum a_n$, if $\lim_{n \to \infty} \sqrt[n]{|a_n|} = L$:
  • If $L < 1$, the series converges absolutely.
  • If $L > 1$, the series diverges.
  • If $L = 1$, the test is inconclusive.

Step 1: Set up the Root Test

Here, we have $a_n = \left( \frac{8n^3 - n - 7}{2n^2 + n + 4} \right)^n$. So, $\sqrt[n]{|a_n|} = \left| \frac{8n^3 - n - 7}{2n^2 + n + 4} \right|.$

Step 2: Evaluate the Limit

To apply the Root Test, find $L = \lim_{n \to \infty} \left| \frac{8n^3 - n - 7}{2n^2 + n + 4} \right|.$

As $n \to \infty$, the dominant terms in the numerator and denominator are $8n^3$ and $2n^2$, respectively. So, $\frac{8n^3 - n - 7}{2n^2 + n + 4} \approx \frac{8n^3}{2n^2} = 4n.$

Thus, $L = \lim_{n \to \infty} 4n = \infty.$

Step 3: Interpret the Result

Since $L = \infty > 1$, the series diverges by the Root Test.

Answer

• Since $L = \infty$, which is greater than 1, the series diverges by the Root Test.

Would you like further details, or do you have any questions?

Related Questions

1. How does the Root Test compare to the Ratio Test for series convergence?
2. What types of series can the Root Test be particularly useful for?
3. Can the Root Test give inconclusive results for some series? If so, in what cases?
4. What are the differences between absolute and conditional convergence?
5. How can polynomial growth in the terms of a series affect convergence?

Tip: Always identify the dominant terms in a function as $n \to \infty$ to simplify limit calculations.