Question: Find the radius of convergence, \( R \), of the series. \[ \sum_{n=1}^{\infty}…

Find the radius of convergence, \( R \), of the series.

\[ \sum_{n=1}^{\infty} \frac{x^{7n}}{n!} \]

\( R = \)

Find the interval, \( I \), of convergence of the series.

\( I = \)

Solution

To find the radius of convergence, \( R \), of the series \[ \sum_{n=1}^{\infty} \frac{x^{7n}}{n!} \] we can use the ratio test. Define \[ a_n = \frac{x^{7n}}{n!} \] Compute the limit \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{|x|^7}{n+1} = 0 \] Since \( 0 < 1 \) for all real numbers \( x \), the radius of convergence \( R \) is \[ R = \infty \] To find the interval of convergence, \( I \), since \( R = \infty \), the interval is all real numbers. \[ I = (-\infty, \infty) \]