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

Authors Avatar

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.

In order to deduce a pattern from the sequences, the varying values for , the sum of the sequences for different values of  will be analyzed. Calculations of the first 10 terms are implemented via GDC and are shown in the following equations. The data is also organized in Table 1.                 (Note: All data in this assignment are correct to six decimal points.)

Join now!

Table 1: Values of  when , ,

In order to visualize the set of data, a graph can be plotted based on the values of .

Figure 1: Value of when  

Observation of both Table 1 and Figure 1 suggest that the value of  increases, approaching 2 as the value of  approaches . With the numbers rounded up to six decimal places, the result of the last three terms round up to the number 2. Thus, a statement about the relationship between the values of  and  can be made using limitation. This is ...

This is a preview of the whole essay