https://file.aiquickdraw.com/mathbot/file/1730170197962gywtk5vf.jpg

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.

Use 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.

Determine the convergence or divergence of the series \( \sum \left( \frac{8n^3 - n - 7}{2n^2 + n + 4} \right)^n \) using the Root Test. Learn step-by-step how to apply the Root Test and interpret the result.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation