Q. 71 - Calculus

Question

A certain power series $\sum _{k=0}^{\infty }{a}_k{x}^k$ has the properties that $a_0$ and $a_1$ are nonzero and $a_{k+2}=a_k$ for every $k>0$. 

(a) Prove that the radius of convergence $\rho$ for the series is finite, and find the value of $\rho$. 

(b) Prove that the series diverges at the endpoints of the interval of convergence. 

(c) What is the sum of the series for $x\in \left(-\rho ,\rho \right)$? Give your answer in terms of $x$, $a_0$, and $a_1$. 

Step-by-Step Solution

Answer

Part a. The radius of convergence for the series is finite and its value is $\rho =1$.

Part b. It is shown that the series diverges at the endpoints of the interval of convergence.   

Part (a) Step 1. Given Information

We are given a power series $\sum _{k=0}^{\infty }{a}_k{x}^k$ having the properties that $a_0,a_1$ are nonzeroes and $a_{k+2}=a_k$ for $k>0$.

Part (a) Step 2. Find the radius of convergence

For the power series $\sum _{k=0}^{\infty }{a}_k{x}^k$ we have