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SL Type 1 PF - Infinite Summation - A general statement has been reached, taking a step further into knowing the infinity.

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Introduction

SL TYPE 1

Portfolio: Infinte Summation

Math SL Section 1

2011-06-18

Infinity has always been the mysterious dark realm of the human kind’s knowledge. Not only is the idea of being able to find out a meaning for infinite sequences of numbers unbelievable, but the process also possesses a level of excitement. In this task, an attempt will be made to discover an exact value for an infinite sequence. Despite the limitations of human—or any other beings—being unable to see the end of the endlessly-continuing series of numbers, steps of mathematical process will be implemented to provide a generalization that is as accurate as possible.

This report will investigate the sum of infinite sequences image00.png, where

image01.png

During the process of obtaining the sum of image00.png, a generalized statement will be reached, based on the pattern of the sequence attained by assessing the influences of image08.png, image05.png, and image04.png on the sequence.

First of all, the assignment provides the values image14.png and image13.png. Applying these values into the initial sequences image00.png, a new sequences is achieved.

image63.png

...read more.

Middle

image17.png8

2.999993

image17.png9

2.999999

image17.png10

3.000000

Table 2: Values of image41.png when image50.png, image51.png, image52.png

image53.png

Figure 2: Value of image41.pngwhen image54.png

Again, observing Table 2 and Figure 2, a noticeable pattern of the numbers is that the values of image09.png approaches 3 as image08.png approaches image48.png. Another generalization using limitation can be made, as denoted following:

image55.png

Now, a general sequence with the image05.png value as 1 will be used to derive a generalized statement.

image56.png

Since calculating an exact value of an undefined number is not possible, an attempt will be made to generalize a statement about image04.png using multiple values for image04.png. Below, in Figure 3, a graph was plotted using the image09.png values when image13.png, image25.png, image26.png, image27.png, image57.png, image58.png, image59.png, image60.png, and image61.png, using the method used previously.

image62.png

Figure 3: Values fo image09.png with multiple image04.png values

Here, depending on the value of image04.png, the values of image09.png approaches relative values of image04.png. When image26.png, image09.png approaches 4 as image08.png approaches image48.png, and when image61.png, image09.png approaches 10 as image08.png approaches image48.png. Without loss of generality, a general statement can be drawn, denoted as following:

image03.png

Now that a general statement for an undefined image04.png is established, the investigation will expand to encompass two undefined numbers, image05.png and image04.png. The sum of image00.png will be determined, where

image06.png

Here, image07.png, the sum of the first image08.png terms for various values of image04.png and image05.png. In order to compare the sums and discover patterns from various image05.png values, the image08.png value will be set at 9 and the image04.png value will be set at 2. Below, Table 3 shows the calculated values for image10.png, using the same method for calculating values of image09.png.

image11.png

image12.png1

2.000000

image12.png2

3.999992

image12.png3

7.999488

image12.png4

15.990193

image12.png5

31.900922

image12.png6

63.331066

image12.png7

124.572949

...read more.

Conclusion

image37.png.

Thus, encompassing all the data and statements derived from previous analyses, a general conclusion can be drawn:

image38.png

There are limitations to this statement. Since the values of all data stretch infinitely when put in their actual form, the values were correct to six decimal places. Thus, at some point as the image08.png values increased, the data showed no difference in the sum of the sequence despite the increase of the image08.png value. For instance, in Table 1, the values of image09.png for image39.png, image40.png, and  showed no visible difference. The expression of using six decimal places also accounts for the decline of the inclination of the data to match the general statement as the image04.png values increased.

Through this assignment, an attempt to discover a general statement about an infinite sequence was undertaken. A general statement has been reached, taking a step further into ‘knowing the infinity.’ Despite the limitations of the precision of number values, the general statement provides a glimpse into the unknown realm of infinity. Mankind has always strived to achieve the ‘impossible.’ This report, along with many great works of numerous mathematicians, shall be an additional proof of mankind’s attempt to tame the infinity.

END

SYU

...read more.

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