To find the radius of convergence and the interval of convergence for the series given by $\frac{x^{n+5}}{2n!}$, we can apply the ratio test.

Step 1: Ratio Test

The general term of the series is:

$a_n = \frac{x^{n+5}}{2n!}$

We apply the ratio test by considering the limit:

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

Calculating $a_{n+1}$:

$a_{n+1} = \frac{x^{(n+1)+5}}{2(n+1)!} = \frac{x^{n+6}}{2(n+1)!}$

Now, calculate $\frac{a_{n+1}}{a_n}$:

$\frac{a_{n+1}}{a_n} = \frac{\frac{x^{n+6}}{2(n+1)!}}{\frac{x^{n+5}}{2n!}} = \frac{x^{n+6}}{x^{n+5}} \cdot \frac{2n!}{2(n+1)!} = x \cdot \frac{1}{n+1}$

Thus, we find:

$L = \lim_{n \to \infty} \left| x \cdot \frac{1}{n+1} \right| = \lim_{n \to \infty} \frac{|x|}{n+1} = 0$

Step 2: Determine Radius of Convergence

Since $L = 0$ for all $x$, the series converges for all $x \in \mathbb{R}$.

Thus, the radius of convergence $R$ is:

$R = \infty$

Step 3: Determine the Interval of Convergence

Since the series converges for all $x$, the interval of convergence is:

$(-\infty, \infty)$

Summary

• Radius of Convergence: $R = \infty$
• Interval of Convergence: $(- \infty, \infty)$

Do you want more details or have any questions about this process? Here are some related questions you might consider:

1. What is the significance of the radius of convergence?
2. How does the ratio test compare with the root test?
3. Can you provide examples of similar series?
4. What happens at the endpoints of convergence in different series?
5. How do factorials influence convergence behavior?

Tip: When analyzing convergence, always consider using multiple tests if you're unsure, especially for complex series.