### Home > APCALC > Chapter 9 > Lesson 9.4.1 > Problem9-113

9-113.

Consider each of the infinite series below. For each series, decide if it converges or diverges and justify your conclusion. If the series converges, calculate its sum. Homework Help ✎

For an infinite geometric series:

$S=\frac{a}{1-r}\text{ }\Big|r\Big|<1$

1. $\frac { 2 } { 3 } + \frac { 1 } { 3 } + \frac { 1 } { 6 } + \frac { 1 } { 12 } + \ldots$

This is a geometric series with $r = 1/2$.

1. $0.9 + 0.99 + 0.999 + 0.9999 + …$

Think of this as:
$0.9 + (0.9 + 0.09) + (0.9 + 0.09 + 0.009) + ...$
What is happening to the value of each term?

1. $0.1 + 0.12 + 0.123 + 0.1234 + …$

See the hint in part (b).

1. $256 − 128 + 64 − 32 + …$

This is a geometric series with $r = -1/2$.