Find radius of convergence of the series ∑∞*𝑛*=0*𝑥*2*𝑛*(2*𝑛*)!.


To find the radius of convergence of the series \(\sum_{n=0}^{\infty} x^{2n} (2n)!\), we will again use the ratio test. 

The general term of the series is:

\[
a_n = x^{2n} (2n)!
\]

Applying the ratio test:

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

where:

\[
a_{n+1} = x^{2(n+1)} (2(n+1))! = x^{2n+2} (2n+2)!
\]

Now calculate the ratio \(\frac{a_{n+1}}{a_n}\):

\[
\frac{a_{n+1}}{a_n} = \frac{x^{2n+2} (2n+2)!}{x^{2n} (2n)!}
\]

Simplify this expression:

\[
\frac{a_{n+1}}{a_n} = x^2 \cdot \frac{(2n+2)(2n+1)(2n)!}{(2n)!} = x^2 (2n+2)(2n+1)
\]

Now, find the limit as \(n \to \infty\):

\[
\lim_{n \to \infty} \left| x^2 (2n+2)(2n+1) \right|
\]

Since \((2n+2)(2n+1) \to \infty\) as \(n \to \infty\), we see that:

\[
\lim_{n \to \infty} \left| x^2 (2n+2)(2n+1) \right| = \infty \text{ if } x \neq 0
\]

Thus, the series diverges for any \(x \neq 0\). The only value of \(x\) for which this series converges is \(x = 0\).

Therefore, the radius of convergence \(R\) is:

\[
R = 0
\]

The series converges only when \(x = 0\).

Would you like further details or have any questions? Here are some related topics you might find interesting:

1. What are the criteria for using the ratio test versus other convergence tests?
2. How does the factorial term affect the convergence of a series?
3. How to find the interval of convergence for a series?
4. What is the significance of the radius of convergence in power series?
5. How does Stirling's approximation help in analyzing factorial terms in series?

**Tip:** When working with factorial terms, consider their rapid growth rate, as they can dominate other terms and significantly influence convergence behavior.

Learn how to find the radius of convergence for the series ∑∞ 𝑛=0 𝑥^(2𝑛) (2𝑛)! using the ratio test. Understand the impact of factorial terms on series convergence. Suitable for advanced undergraduate students and those interested in series convergence.

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