Improper Integrals and Sums of Series - Assignment Example

Improper Integrals 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 rain drops 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. Descending to an earthlier plain, we consider the situation posed by the Ozimandus Corporation who has decided to market the Deluxe set by painting them over with a highly expensive platinum gild. For those uninitiated to this discussion, The Deluxe set is a product range from the Blocks Unlimited Store that features an infinite number of cubes such that they have sizes that are arranged in a harmonic progression of natural numbers. The Corporation is interested in optimizing the production costs and are 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$. Subsequently it remains to be seen what is the required surface area that has to be painted. In the succeeding discussion, the details of the solution are given. For most of the discussion, it shall be assumed that the reader is conversant with basic calculus (particularly integration). Some particular ideas are easier and only elementary algebra is considered a requisite. It is well known from elementary algebra that a cube of size ‘a’ has a volume given by a3 and a total surface area of 6a2. Thus, the surface area of the first cube of the Deluxe Set (DS from here onwards) is simply 6 sq.ft while that of the second cube is sq.ft. The total surface that is compared by DS is exactly given by: For convenience we denote the infinite series so generated by the Greek letter ξ. Our task now is to determine the value of ξ; or if this is not possible then at least to approximate ξ. It had been mentioned in the previous essay that . Thus as a starting approximation we hold that the total surface area covered by the DS is less than 12 square foot and thereby the total cost is in any case less than 120$. As we shall see, much closer approximations are possible. Consider the graph of the function  : Note that the area of all the smaller rectangles are the same as the series we require . From the graph it is easy to understand that the area under the curve is more than the sum of all the individual rectangles. Mathematically, From elementary integral calculus, we know that the area under the curve is given by .  or starting from an arbitrary integer m, . Adding the first m terms of ξ to this, we have that It can be easily shown that . Finally, Despite the intimidating looking formula, it requires only a pocket calculator to be computed. Starting with m=3, we get . This gives a surface area less than 10.2 square feet. Taking this two steps further and setting m=5, we get that . This takes the estimate even lesser than 10 sq.ft to 9.89. Thus, we see now that total cost of painting the DS is lesser than even 100$. But can it be even lesser than 90$? Unfortunately, this cannot be so, as shall be demonstrated below. The areas of the first seven boxes of the deluxe set are as follows: 6, 1.5, 0.66, 0.375, 0.24, 0.1668 and 0.1224. Adding these we get the total surface areas of only the five boxes to be 9.0642 sq ft. Thus it requires 90.64$ just to paint the first seven boxes and there is no way that the entire set can be painted at a rate less than this. The exact value of ξ has been found to be. This corresponds to a surface area of 9.86 sq ft or the minimum price has to be at least 98.6$. Having settled the problems at Ozimandus Corporation, we turn our attention to a query from an aging motorcyclist, Joe Ventura, who wishes to know if he can take away the entire Deluxe Set in the back of his antique Messerschmitt automobile. The boot of the car has the dimensions of 1x1x1.25 ft. Thus the effective boot space of the Messerschmitt is 1.25 cubic feet. Is this more than the total volume of the Deluxe Set? We proceed as in the previous example; however, this time we take the volumes instead of the area. The total volume of the Deluxe Set therefore is: From the graph of , we find that  or equivalently, . Adding the first m terms of ξ to this, we have that Or, . Taking the very first case for m=2, we get, Thus we see β or the volume of the Deluxe set is indeed less than the boot space of the Messerschmitt. Theoretically, the car, despite its mouse version, should be able to contain an infinite number of solid cubes. But this does not tell us if this is practically possible. For that we have to see if the dimensions of the boot space permit the Deluxe set to be so contained. The boot is characterized as a 1x1x1.25 volume enclosure. However, the first cube of the Deluxe set has a dimension of 1x1x1 ft. Thus, over 80% of the space is occupied by the very first cube leaving only a small space at the top for all the other boxes. The dimensions of this free space is 1x1x0.25. That is, there is only a quarter feet left at the top of the boot for all the other boxes. But the second box has a length of half a feet or 0.5ft! Therefore, we are left with the awkward situation that even though there is enough space in the boot for an infinite number of boxes, the dimensional constraint is such that even two boxes cannot be squeezed in! The paradoxes of infinity, once again in action! Thus we conclude another round of interesting discussions regarding infinite series. In each case, it was the tying up of an infinite yet discrete series to an integral of a continuous function that enables us the ease of computation which we otherwise might not have had. This is an important mathematical concept, whose significance we missed only because we approached the problem geometrically and not analytically. Geometry permits understanding and exploration of ideas that are otherwise invisible to the untrained eye. The recognition of this fact lies at the heart of all coordinate geometry and analysis. Bibliography Erickson, G W and J A Fossa. Dictionary of Paradox. New York: Univeristy of America Press, 1998. Rudin, Walter. Principles of Mathematical Analysis. McGraw Hil Publishing, 1976. Read More