Chapter 10.4, Problem 32E

To determine

To find: the convergence or divergence of the series.

Answer to Problem 32E

Converges using the n^th term test

Explanation of Solution

Given:

  ${\displaystyle \sum _{n=1}^{\infty }-\frac{1}{8^n}}$

Calculation:

Using n^th term test

  $\underset{n\to \infty }{\lim }{a}_n$

  $\underset{n\to \infty }{\lim }-\frac{1}{8^n}$

  $\frac{1}{\infty }=0$

  $\underset{n\to \infty }{\lim }-\frac{a_{n+1}}{a_n}$

  $\underset{n\to \infty }{\lim }\frac{-8^n}{-8^{n+1}}$

  $\frac{x^{n+1}}{x^n}=x$

  $\frac{1}{8}<1$

Converges