do an example of interval of convergnece with root test

### Example: Interval of Convergence Using the Root Test

Consider the series:  
\[
\sum_{n=1}^\infty \frac{x^n}{n^n}.
\]

We will determine its interval of convergence using the **Root Test**.

---

#### Step 1: Write the general term of the series
The \(n\)-th term of the series is:  
\[
a_n = \frac{x^n}{n^n}.
\]

---

#### Step 2: Apply the Root Test
The Root Test states that for the series \(\sum a_n\), compute:  
\[
\lim_{n \to \infty} \sqrt[n]{|a_n|}.
\]
1. If the limit \(L < 1\), the series converges absolutely.
2. If \(L > 1\), the series diverges.
3. If \(L = 1\), the test is inconclusive.

For \(a_n = \frac{x^n}{n^n}\):  
\[
|a_n| = \frac{|x|^n}{n^n}.
\]  
Now compute:  
\[
\sqrt[n]{|a_n|} = \sqrt[n]{\frac{|x|^n}{n^n}}.
\]

---

#### Step 3: Simplify the Root Expression
Using the property of roots, we get:  
\[
\sqrt[n]{|a_n|} = \frac{\sqrt[n]{|x|^n}}{\sqrt[n]{n^n}}.
\]  

1. Simplify \(\sqrt[n]{|x|^n}\):  
   \[
   \sqrt[n]{|x|^n} = |x|.
   \]
2. Simplify \(\sqrt[n]{n^n}\):  
   \[
   \sqrt[n]{n^n} = n.
   \]

Thus,  
\[
\sqrt[n]{|a_n|} = \frac{|x|}{n}.
\]

---

#### Step 4: Take the Limit as \(n \to \infty\)
\[
\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \frac{|x|}{n} = 0.
\]

---

#### Step 5: Determine Convergence
Since the limit is \(0 < 1\) for all \(x\), the series **converges for all \(x\)**.

---

#### Final Answer:  
The series converges **for all real numbers \(x\)**. The **interval of convergence** is:
\[
(-\infty, \infty).
\]

---

Do you want more details or have any questions? Here are 5 related questions to expand on this topic:

1. What is the difference between the Root Test and the Ratio Test for determining convergence?  
2. Can you demonstrate finding the radius of convergence for a power series using the Root Test?  
3. How does absolute convergence relate to the Root Test?  
4. How would the result change if the \(n^n\) term in the denominator were replaced by \(n!\)?  
5. Can the Root Test ever fail to determine convergence? Provide an example.

**Tip:** Always simplify the general term as much as possible before applying the Root Test for easier computation!

do an example of interval of convergence with root test

Understand how to determine the interval of convergence for the series Σ(x^n / n^n) using the Root Test. This step-by-step example simplifies the root test process, computes the limit, and concludes with the series converging for all x. Ideal for high school and college students.

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