# prepare and submit a paper on improper integrals and sums of series.

Your assignment is to prepare and submit a paper on improper integrals and sums of series.

The paradoxes that we face when we encounter infinity had been considered in some detail in the previous paper. The idea that the sum of an infinite number of terms can sometimes summate to infinity while at other times sum up to a finite limit is contrary to common sense. From our experience, we are aware of how an (apparently) infinite number of raindrops add up to give gigantic oceans and how microscopic gradation of wind erodes away entire mountain ranges. On closer contemplation, we can see that such common concepts as “the power of the masses”, “an army of ants” etc all carry with them the inherent belief that an infinite number of members leads to infinite strength.

The Corporation is interested in optimizing the production costs and is in a stage of consultation with industry experts. Specifically, they wish to maintain the usage of the expensive platinum gild to the minimum as this is unsurprisingly the most expensive part of the production process. It has been informed that the cost of one square foot of the platinum gild material has been contained to 10$. Subsequently, it remains to be seen what is the required surface area that has to be painted. In the succeeding discussion, the details of the solution are given. For most of the discussion, it shall be assumed that the reader is conversant with basic calculus (particularly integration). Some particular ideas are easier and only elementary algebra is considered a requisite.

It is well known from elementary algebra that a cube of size ‘a’ has a volume given by a3 and a total surface area of 6a2. Thus, the surface area of the first cube of the Deluxe Set (DS from here onwards) is simply 6 sq.ft while that of the second cube is The total surface that is compared by DS is exactly given by:

For convenience, we denote the infinite series so generated by the Greek letter ξ. Our task now is to determine the value of ξ. or if this is not possible then at least to approximate ξ. It had been mentioned in the previous essay. Thus, the surface area of the first cube of the Deluxe Set (DS from here onwards) is simply 6 sq.ft while that of the second cube is sq.ft. .