Visualizing Euler's Constant

n

7

Euler's constant is

γ=lim

n∞

1+

1

2

+

1

3

+...+

1

n

-ln(n+1)

.

For each

n

,

1+

1

2

+

1

3

+⋯+

1

n

is the sum of the areas of the rectangles in the graph, and

ln(n+1)

is the area under the curve

y=

1

x

over the closed interval

[1,n+1]

. Therefore

γ

is the area of the green region.

Reference: D. W. Temple, "A Geometric Look at Sequences that Converge to Euler's Constant," The College Mathematics Journal, 37(2), 2006 pp. 128–131.

Soledad Mª Sáez Martínez, Félix Martínez de la Rosa