Question: Find the radius of convergence, \( R \), of the series. \[ \sum_{n=3}^{\infty} \frac{x^n +…

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

\[ \sum_{n=3}^{\infty} \frac{x^n + 1}{5n!} \]

\( R = \boxed{\phantom{0}} \)

Find the interval, \( I \), of convergence of the series. (Enter your answer using interval notation.)

\( I = \boxed{\phantom{0}} \)

Solution

Step 1: Consider the given series \[ \sum_{n=3}^{\infty} \frac{x^n + 1}{5n!} \] Step 2: To find the radius of convergence \( R \), we apply the Ratio Test. Step 3: Compute the ratio of consecutive terms \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \left| \frac{x^{n+1} + 1}{5(n+1)!} \cdot \frac{5n!}{x^n + 1} \right| \] Step 4: Simplify the expression \[ \lim_{n \to \infty} \left| \frac{x^{n+1} + 1}{(n+1)(x^n + 1)} \right| \] Step 5: As \( n \) approaches infinity, the dominant term in the numerator and denominator is \( x^{n+1} \) and \( x^n \) respectively. Therefore, the limit becomes \[ \lim_{n \to \infty} \left| \frac{x \cdot x^n}{(n+1) \cdot x^n} \right| = \lim_{n \to \infty} \left| \frac{x}{n+1} \right| = 0 \] Step 6: Since the limit is \( 0 \) and \( 0 < 1 \), the series converges for all real numbers \( x \). Step 7: Therefore, the radius of convergence \( R \) is \[ R = \infty \] Step 8: The interval of convergence \( I \) is all real numbers \[ I = (-\infty, \infty) \]