Question: (a) Use differentiation to find a power series representation for \[ f(x) = \frac{1}{(5 + x)^2}…

(a) Use differentiation to find a power series representation for \[ f(x) = \frac{1}{(5 + x)^2}. \] \[ f(x) = \sum_{n=0}^{\infty} \left( \frac{(-1)^n (n + 1) x^n}{5^{n+2}} \right) \] What is the radius of convergence, \( R \)? \[ R = 10 \]

(b) Use part (a) to find a power series for \[ f(x) = \frac{1}{(5 + x)^3}. \] \[ f(x) = \sum_{n=0}^{\infty} \left( \frac{(-1)^n (n + 1) (n + 2) x^n}{2(5^{n+3})} \right) \] What is the radius of convergence, \( R \)? \[ R = \boxed{\phantom{10}} \]

(c) Use part (b) to find a power series for \[ f(x) = \frac{x^2}{(5 + x)^3}. \] \[ f(x) = \sum_{n=2}^{\infty} \left( \frac{(-1)^n n (n - 1) x^n}{2(5^{n+1})} \right) \] What is the radius of convergence, \( R \)? \[ R = \boxed{\phantom{10}} \]

Solution

Part (a): Find the Radius of Convergence for \( f(x) = \frac{1}{(5 + x)^2} \) Step 1: Rewrite the function to resemble the standard geometric series. \[ f(x) = \frac{1}{5^2 \left(1 + \frac{x}{5}\right)^2} & \] Step 2: Recognize that the power series converges when the absolute value of the ratio is less than 1. \[ \left| \frac{x}{5} \right| < 1 & \] Step 3: Solve the inequality to find the radius of convergence. \[ |x| < 5 & \] Radius of Convergence, \( R = 5 \) --- Part (b): Find the Radius of Convergence for \( f(x) = \frac{1}{(5 + x)^3} \) Step 1: Rewrite the function to resemble the standard geometric series. \[ f(x) = \frac{1}{5^3 \left(1 + \frac{x}{5}\right)^3} & \] Step 2: Identify the condition for convergence based on the ratio. \[ \left| \frac{x}{5} \right| < 1 & \] Step 3: Solve the inequality to determine the radius of convergence. \[ |x| < 5 & \] Radius of Convergence, \( R = 5 \) --- Part (c): Find the Radius of Convergence for \( f(x) = \frac{x^2}{(5 + x)^3} \) Step 1: Rewrite the function to resemble the standard geometric series. \[ f(x) = \frac{x^2}{5^3 \left(1 + \frac{x}{5}\right)^3} & \] Step 2: Recognize that the multiplication by \( x^2 \) does not affect the radius of convergence. Step 3: Identify the condition for convergence based on the ratio. \[ \left| \frac{x}{5} \right| < 1 & \] Step 4: Solve the inequality to find the radius of convergence. \[ |x| < 5 & \] Radius of Convergence, \( R = 5 \)