# Area sum of inscribed pentagons

1. Aug 15, 2011

1. The problem statement, all variables and given/known data

Ok, I wanted to calculate the sums of the areas of inscribed petagons into an initial pentagon (the smaller pentagon's vertices touch the larger pentagon's midpoints). However, I wanted all of the even pentagons to create negative space, and then the odd # pentagon after that to create positive space.

The initial pentagon is regular with side length of 1. To calculate the area of the smaller pentagons, i assumed that the area would be proportional to the next bigger pentagon by a constant factor. apothem/radius

i.e. area of small pentagon = area of big pentagon *(apothem/radius)

Total area = area1 - area2 + area3 - area4 + area5 - ...

So to do this I calculated a sum (see picture). Can someone verify that my sum is correct?

My question: How does my calculator take an infinite sum and convert it into a nice and clean polynomial? Can this always be done? What conditions need to be met to do so (obviously the sum need to converge, but are there any other requirements?)

2. Relevant equations

http://img.photobucket.com/albums/v298/Swiffer/Math.jpg [Broken] Sorry for the blur, my phone can only do 3 megapixel resolution.

3. The attempt at a solution

See picture.