### Home > CALC > Chapter 9 > Lesson 9.4.1 > Problem9-114

9-114.

Consider the infinite series below. For each, decide if it converges or diverges and justify your conclusion. If the series converges, find its sum.

For an infinite geometric series:

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

This is a geometric series with $r =\frac{1}{2}$.

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 $r =-\frac{1}{2}$.