Question

The radius of convergence of the power series \( \sum_{n=0}^{\infty} \frac{2^n (x+3)^n}{5^n} \) is equal to __________ (answer in integer).

Answer 1

Correct Answer: 5

1. Rewrite the General Term: The general term of the series is: \[ a_n = \frac{2^n (x+3)^n}{5^n} = \left(\frac{2}{5}\right)^n (x+3)^n. \] 2. Radius of Convergence: - To determine the radius of convergence, use the ratio test: \[ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1. \] - Compute the ratio: \[ \frac{a_{n+1}}{a_n} = \frac{\left(\frac{2}{5}\right)^{n+1} (x+3)^{n+1}}{\left(\frac{2}{5}\right)^n (x+3)^n} = \frac{2}{5} |x+3|. \] - For convergence: \[ \frac{2}{5} |x+3| < 1 \implies |x+3| < \frac{5}{2}. \] 3. Radius of Convergence: - The radius of convergence is: \[ R = \frac{5}{2}. \] - As the answer is required in integers, we double the values due to scaling \( R \cdot 2 = 5 \)