Chapter 9, Problem 89RE

(a)

To determine

The interval of convergence of the provided power series $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\left(\frac{x}{5}\right)}^n}$.

(b)

To determine

The interval of convergence of the derivative of the function $f\text{'}\left(x\right)$ where the function is $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\left(\frac{x}{5}\right)}^n}$.

(c)

To determine

The interval of convergence of the $f\text{'}\text{'}\left(x\right)$ where $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\left(\frac{x}{5}\right)}^n}$.

(d)

To determine

The interval of convergence of ${\displaystyle \int f\left(x\right)}$ where $f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\left(\frac{x}{5}\right)}^n}$.