https://file.aiquickdraw.com/mathbot/file/1726954229294qm0mlxws.jpg

To find the interval of convergence for the given series:

\[
\sum_{n=1}^{\infty} \frac{(-11x)^n}{\sqrt{n+7}}
\]

We need to follow these steps:

### Step 1: Apply the Ratio Test
The ratio test helps us determine where the series converges. Given the series' general term:
\[
a_n = \frac{(-11x)^n}{\sqrt{n+7}}
\]
The ratio test considers the limit:
\[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
\]

Let’s calculate this ratio:

\[
\frac{a_{n+1}}{a_n} = \frac{\frac{(-11x)^{n+1}}{\sqrt{(n+1)+7}}}{\frac{(-11x)^n}{\sqrt{n+7}}} = \frac{(-11x)^{n+1}}{(-11x)^n} \cdot \frac{\sqrt{n+7}}{\sqrt{n+1+7}}
\]

Simplifying:
\[
= (-11x) \cdot \frac{\sqrt{n+7}}{\sqrt{n+8}}
\]

Now take the absolute value:
\[
\left| \frac{a_{n+1}}{a_n} \right| = 11|x| \cdot \frac{\sqrt{n+7}}{\sqrt{n+8}}
\]

As \(n \to \infty\), \(\frac{\sqrt{n+7}}{\sqrt{n+8}} \to 1\), so the limit becomes:
\[
L = 11|x|
\]

### Step 2: Apply the Ratio Test Criterion
For the series to converge, we require:
\[
L = 11|x| < 1
\]
Thus, the series converges when:
\[
|x| < \frac{1}{11}
\]

### Step 3: Determine the Radius and Interval of Convergence
The radius of convergence is:
\[
R = \frac{1}{11}
\]
Thus, the interval of convergence is:
\[
-\frac{1}{11} < x < \frac{1}{11}
\]

### Step 4: Check the Endpoints
We now test the endpoints \(x = \pm \frac{1}{11}\).

- For \(x = \frac{1}{11}\), the series becomes:
  \[
  \sum_{n=1}^{\infty} \frac{(-11 \cdot \frac{1}{11})^n}{\sqrt{n+7}} = \sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n+7}}
  \]
  This is an alternating series, and since \(\frac{1}{\sqrt{n+7}}\) decreases and approaches 0, it converges by the Alternating Series Test.

- For \(x = -\frac{1}{11}\), the series becomes:
  \[
  \sum_{n=1}^{\infty} \frac{(-11 \cdot \left(-\frac{1}{11}\right))^n}{\sqrt{n+7}} = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n+7}}
  \]
  This is a positive term series that behaves like the harmonic series, which diverges.

### Final Answer:
The interval of convergence is:
\[
\left[-\frac{1}{11}, \frac{1}{11}\right)
\]
- The left endpoint \(x = -\frac{1}{11}\) is **included** (Y).
- The right endpoint \(x = \frac{1}{11}\) is **not included** (N).

Would you like further details or explanations?

### 5 Related Questions:
1. What is the difference between conditional and absolute convergence?
2. How does the Alternating Series Test work for endpoint evaluation?
3. Can we use the root test instead of the ratio test for this series?
4. Why does the series diverge at \(x = -\frac{1}{11}\)?
5. How does changing the coefficient of \(x\) affect the interval of convergence?

### Tip:
When checking for convergence at endpoints, always test for both conditional and absolute convergence to ensure accuracy.

Find the interval of convergence of the following series: 
∑_(n=1)^∞ ((-11x)^n) / (√(n + 7))

Learn how to find the interval of convergence for the series ∑ ((-11x)^n) / (√(n + 7)). This calculus problem involves applying the ratio test and the alternating series test. It covers the step-by-step process of determining the radius and interval of convergence, as well as testing the endpoints for convergence.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation