Question

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. (If the series is divergent, enter DIVERGENT.)

1- 1/6 + 1/36 - 1/216 + . . .

Answer 1

\[\begin{align*}1-\frac{1}{6}+\frac{1}{36}-\frac{1}{216}+\cdots&=\left(-\frac{1}{6}\right)^0+\left(-\frac{1}{6}\right)^1+\left(-\frac{1}{6}\right)^2+\left(-\frac{1}{6}\right)^6+\cdots\\
&=\sum_{n=0}^\infty \left(-\frac{1}{6}\right)^n
\end{align*}\]

Answer 2

How can you tell whether a geometric series converges?

Answer 3

So what would the sum be?

Answer 4

we use a formula to find the sum of a geometric series

Answer 5

sum = a / (1-r), where a = first term. r = ratio . here a = 1 , r = -1/6

Answer 6

1/ ( 1 - (-1/6) = 1 / ( 1 + 1/6) = 1/ ( 7/6) = 6/7