Chapter 9.8, Problem 68E

(a)

To determine

To calculate: The sum of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(3x\right)}^n}$ at the point $x=\frac{1}{6}$ when the geometric series converges on $\left(-\frac{1}{3},\frac{1}{3}\right)$.

(b)

To determine

To calculate: The sum of the series ${\displaystyle \sum _{n=0}^{\infty }{\left(3x\right)}^n}$ when $x=-\frac{1}{6}$ as the geometric series converges on $\left(-\frac{1}{3},\frac{1}{3}\right)$.

(c)

To determine

The comparison between the rates of convergence of the partial sums with the sums of the series in parts (a) and (b).

(d)

To determine

To calculate: The value of the positive integer N for given M.