Sum of a Telescoping Series

n

2

For each

n=1,2,…

, the area of the corresponding colored region is

1

n

-

1

n+1

=

1

n(n+1)

, therefore

∞

∑

n=1

1

n(n+1)

=1

.

Reference: Füsun Akman, "Proofs without Words under the Magic Curve," The College Mathematics Journal, 33(1), 2002 pp. 42–46.

Soledad Mª Sáez Martínez, Félix Martínez de la Rosa