Chapter 8.5, Problem 33E

To determine

To find: The power series and interval of convergence .

Answer to Problem 33E

Hence, the radius of the convergence is $R=2$ .

Explanation of Solution

Given information: ${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ and ${\displaystyle \sum _{n=0}^{\infty }{b}_n}{x}^n$ .

If $R_1$ and $R_2$ be the radius of convergence of the power series ${\displaystyle \sum _{n=0}^{\infty }{a}_n{x}^n}$ and ${\displaystyle \sum _{n=0}^{\infty }{b}_n}{x}^n$ then the radius of convergence of ${\displaystyle \sum _{n=0}^{\infty }(a_n}+b_n)x^n$is $R=\min \left\{R_1,R_2\right\}$

Hence, the radius of the convergence of ${\displaystyle \sum _{n=0}^{\infty }(a_n}+b_n)x^n$is $R=\min \left\{2,3\right\}$

  $R=2$