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

Extracts from this document...

Introduction

 SL TYPE 1 Portfolio: Infinte Summation Math SL Section 12011-06-18

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 , where

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

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

Middle

8

2.999993

9

2.999999

10

3.000000

Table 2: Values of  when , ,

Figure 2: Value of when

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

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

Since calculating an exact value of an undefined number is not possible, an attempt will be made to generalize a statement about  using multiple values for . Below, in Figure 3, a graph was plotted using the  values when , , , , , , , , and , using the method used previously.

Figure 3: Values fo  with multiple  values

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

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

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

 1 2.000000 2 3.999992 3 7.999488 4 15.990193 5 31.900922 6 63.331066 7 124.572949

Conclusion

.

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

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  values increased, the data showed no difference in the sum of the sequence despite the increase of the  value. For instance, in Table 1, the values of  for , , 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  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

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