WOLFRAM|DEMONSTRATIONS PROJECT

A Converging Geometric Series

n

3

area = 0.875000000

This Demonstration shows that

∞

∑

n=1

n

1

2

=

1

2

+

1

4

+

1

8

+…+

n

1

2

=1

.

The square is half empty with area

1

2

when

n=1

. The square is

1

4

empty at

n=2

. In general, the square is

n

1

2

empty at step

n

and

1

2

+

1

4

+

1

8

+…+

n

1

2

full, which shows that

∞

∑

n=1

n

1

2

converges to 1.