Sum of a Telescoping Series (II)

n

6

The area of each colored region is

n

n+1

-

n-1

n

=

1

n(n+1)

, so

1

1·2

+

1

2·3

+⋯+

1

n(n+1)

=

n

n+1

and

∞

∑

n=1

1

n(n+1)

=1

.

Details

Reference: R. W. Wang, "Proof without Words:

1

1·2

+

1

2·3

+⋯+

1

n(n+1)

=

n

n+1

," Mathematics Magazine, 65(5), 1992 p. 338.

External Links

Series (Wolfram MathWorld)

Permanent Citation

Soledad Mª Sáez Martínez, Félix Martínez de la Rosa

"Sum of a Telescoping Series (II)"
http://demonstrations.wolfram.com/SumOfATelescopingSeriesII/
Wolfram Demonstrations Project
Published: March 7, 2011