Chapter 9.5, Problem 22PPSE

Solution

To find whether the series $-54-18-6-\dots$ converges or diverges, and find the sum if the converges.

The series converges, and the sum is $-81$ .

Given information:

The series $-54-18-6-\dots$ .

Calculation:

For the given geometric series, $a_1=-54$ and $r=\frac{1}{3}$ .

Since $\left|r\right|=\left|\frac{1}{3}\right|=\frac{1}{3}<1$ , the series converges.

Find the sum of the series using the formula, $S_{\infty }=\frac{a_1}{1-r}$ .

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{-54}{1-\frac{1}{3}}\\ =-54\cdot \frac{3}{2}\\ =-81\end{array}$

Conclusion:

The series converges, and the sum is $-81$ .