The variables

,

,

,

are positive integers and

is real and non-negative.

This Demonstration plots the curve

over the interval

,

. The vertical lines at

intersect the curve and the line from

to

at red points; those points form nonoverlapping polygons. The regions between the curve and the secants joining the red points on the curve can be partially filled by polygons, too; this is the second layer. In general, the area of each polygon formed at layer

at position

on the interval

is computed and approximated.

The area

of the polygon with end vertices on the curve at

and

is precisely the sum of the areas of the yellow and the gray triangles with red dots at vertices shown in the diagram, so the expression for the area is

.

For example,

.

The limit

as

gives the closed area above the curve, and is in principle the method of Riemann integration.

The area

of the polygon with end vertices on the curve at

and

at layer

at position

on the interval

is

,

, with value shown in the diagram as "area of moving polygon".

Considering the polygons as a series of "layers" is key to the following algebra.

Re-indexing

as used in the diagram plot rewrites the

^{th} term of the infinite sequence of polygon areas in a layer

as

, giving

.

The sum of the first layer of polygon areas on the infinite curve is then

and the sum of all polygon layers is

.

This can be simplified to

.

Rearranging the order of summation into tuples in order of size produces the single sum over

, which is easier to compute:

.

Looking at the dissection diagram and its algebraic expression lets us see many polygon dissections for given closed areas.

Putting

in the diagram plots the hyperbola

and we observe a sum of polygon areas giving the identity

Putting

we observe

for

.

The term

is the area under the curve for

.

The term

is the total area of the right triangles above the curve at positive integer points, and by telescoping is constant for

.

The term

is the total area of all polygons on the infinite curve.

At the special value

,

gives the Euler–Mascheroni constant

(see [2]).

The sum

.

Rewriting

shows the Vacca-type rational series for the Euler–Masceroni constant

due to Sondow [3, 4].

for

a positive integer greater than 1 [3].

Putting

, we observe

for

.

The term

alternates the areas under the curve. Writing the integrals gives the term as

,

with the factor

excising the pole in the expression when

and the term value is

;

and

are the ceiling and floor functions, respectively.

The term

is the alternating sum of the right triangle areas above the curve at positive integer points.

The term

alternates the summation of the areas of all the polygons on the curve.

At the special value

,

gives the constant

(see [3, 4]);

relates to a problem mentioned in [4].