Find the radius of convergence of Summation from nothing to nothing StartFraction 2 k exclamation mark x Superscript 4 k Over left parenthesis 4 k right parenthesis Superscript k EndFraction
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Part 1
The radius of convergence is 
  
enter your response here.
​(Type an exact answer in terms of e​.)

To find the radius of convergence of the series

\[
\sum_{k=0}^{\infty} \frac{2k! \, x^{4k}}{(4k)^k}
\]

we can use the **ratio test**, which states that for a series \(\sum a_k x^k\), the radius of convergence \(R\) can be found as:

\[
\frac{1}{R} = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|
\]

For this series, let’s focus on the term:

\[
a_k = \frac{2k!}{(4k)^k} x^{4k}
\]

The key is to examine how \(a_k\) behaves as \(k \to \infty\) by isolating the dependency on \(x\).

### Step-by-Step Solution

1. **Set Up the Ratio of Consecutive Terms**:
   \[
   \frac{a_{k+1}}{a_k} = \frac{\frac{2(k+1)!}{(4(k+1))^{k+1}} x^{4(k+1)}}{\frac{2k!}{(4k)^k} x^{4k}}
   \]

2. **Simplify the Ratio**:
   Since this problem’s complexity suggests direct simplification may be challenging, we can alternatively consider **root convergence** methods due to the structure of \(x^{4k}\) term.



Find the radius of convergence of Summation from nothing to nothing StartFraction 2 k exclamation mark x Superscript 4 k Over left parenthesis 4 k right parenthesis Superscript k EndFraction.

Explore how to calculate the radius of convergence for the series \(\sum_{k=0}^{\infty} \frac{2k! \, x^{4k}}{(4k)^k}\) using the ratio and root tests. Learn key concepts like radius of convergence and advanced series analysis, suitable for undergraduate students.

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