In 1735 Euler famously proved that
His method was purely algebraic, as are many of the multitude of proofs that appear for this result. However, various geometric perspectives exist, and they may illuminate some of the properties of .
As previously discussed, we may represent by the double integral
The symmetry of this representation allows us calculate the exact value of . See the previous post for details of this.
Another geometric representation of is as the area between the curve and the positive -axis.
Or formulated as an integral
To show that the above integral is indeed equal to we consider the Mercator series (expansion of the natural logarithm)
Interchanging the order of summation and integration may be justified as is a sequence of positive integrable functions (see e.g here). What follows is a geometric representation for the individual terms of , through the equation
as originally claimed.
Another way to see that the area under the curve is (attributed to Johan Wästlund) is given in this math.SE post. Indeed most of what follows is tracing through the comments there for my own personal edification.
We construct a geometric object in the positive quadrant of over iterations . For example, is shown below. (Image by Hans Lundmark)
We let be the square . Then is obtained from by removing the ‘outermost’ diagonal layer of rectangles and adding two sets of rectangles of area and a square of area to cover the ‘inner edges’.
We give a visual representation of this process going from to whereby in constructing from we remove red rectangles and add green ones.
It may be observed that each rectangle removed is equal to two of the rectangles added, namely the th rectangle from the top and the th rectangle from the left of those added has area equal to the th rectangle on the diagonal.
Therefore the area added at stage in the process above is only that of the square . Thus the area of is and thus in the limit the area of approaches .