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.

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Step 1: Write the general term of the series

The $n$-th term of the series is:
$a_n = \frac{x^n}{n^n}.$

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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}}. ]

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

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Step 4: Take the Limit as $n \to \infty$

$\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \frac{|x|}{n} = 0.$

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Step 5: Determine Convergence

Since the limit is $0 < 1$ for all $x$, the series converges for all $x$.

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Final Answer:

The series converges for all real numbers $x$. The interval of convergence is: $(-\infty, \infty).$

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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!