Concept Byte: Geometry and Infinite Series

For Use With Lesson 9-5

You can use geometric figures to model some infinite series.

Activity 1

Geometry Draw a geometric figure to model the series.

1 2 + ( 1 2 ) 2 + ( 1 2 ) 3 + ⋯ + ( 1 2 ) n + … 1 half , plus . open , 1 half , close squared . plus . open , 1 half , close cubed . plus math axis ellipsis plus . open , 1 half , close to the n . plus dot dot dot

Image Long Description

Draw a square. Shade one half of the square. Then shade one half of the remaining unshaded region. Continue until the square is full.

So the series appears to have a sum of 1.

You can write an infinite series from a geometric model.

Activity 2

Geometry Write the series modeled by the trapezoids. Estimate the sum of the series. Explain your reasoning.

Image Long Description

The shaded region approaches one third of the figure.

So the series 1 4 + ( 1 4 ) 2 + ( 1 4 ) 3 + ⋯ + ( 1 4 ) n + … 1 fourth , plus . open , 1 fourth , close squared . plus . open , 1 fourth , close cubed . plus math axis ellipsis plus . open , 1 fourth , close to the n . plus dot dot dot  appears to have a sum of 1 3 . 1 third , .

Exercises

1. 1. Write the series modeled by the figure below.
   2. Evaluate the series. Explain your reasoning.

   
   Image Long Description

2. Draw a figure to model the series.

   1 5 + ( 1 5 ) 2 + ( 1 5 ) 3 + ⋯ + ( 1 5 ) n + … 1 fifth , plus . open , 1 fifth , close squared . plus . open , 1 fifth , close cubed . plus math axis ellipsis plus . open , 1 fifth , close to the n . plus dot dot dot

3. Make a Conjecture Consider the series.

   1 c + ( 1 c ) 2 + ( 1 c ) 3 + ⋯ + ( 1 c ) n + … , c > 1 1 over c , plus . open , 1 over c , close squared . plus . open , 1 over c , close cubed . plus math axis ellipsis plus . open , 1 over c , close to the n . plus dot dot dot comma c greater than 1

   What is the sum of the series? Explain your reasoning.

Table of Contents