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# Infinite Summation Portfolio. I will consider the general sequence with constant values

Extracts from this document...

Introduction

IB Mathematics SL Type 1: Infinite Summation Portfolio

The English School

IB Mathematics SL

Math Portfolio (Type1)

Infinite Summation

Candidate Code:000185

19th of October 2011

Bogotá, Colombia

Aim: The aim of this portfolio is to research about the summation of infinite series. In first place I will consider the general sequence with constant values for  and variables for . The , which is calculate the sum of the first terms of the above sequence for  will be calculated with the programme of Microsoft Excel and illustrated in graphs with Graphmatica; another computer based program, so that we can create a general statement which can be proven. Furthermore, to expand the investigation I will explore the same general sequence, but with variables in both  and . I will accomplish this by doing the same method as the first exercises. Finally, I will conclude by showing the scopes/limitations of the general statement.

Where,

, ,  ,   ,

We must take into account that factorial notation is in succession, this means that the factorial notation shows all the natural numbers from 1 to , in such way that:

Following the first exercise, we will change the variables of terms and , in such way that and . If we replace these values into the original sequence we obtain:

, ,  ,  , …

In order to calculate the sum of the first terms of the above sequence for  we must know that the sum of the first 10 values is in progression. Hence, I will use Microsoft Excel in order to plot results in a suitable table. The first column will contain the different values of

Middle

, likewise the way  approaches to .

Next, we will consider the same general sequence where  in such way that:

,   ,   ,    ,

In this sequence we will replace different values for variable . In my case I will use 3 different values that can replace : 4, 5 and 6. For each of them I will calculate the sum of the first terms of the above sequence for . To do this I will use again the help of Microsoft Excel as done previously in other sequences.

 Replacing in the form 0 1 1 1 1,38629436 2,38629436 2 0,96090603 3,34720039 3 0,44403287 3,79123326 4 0,15389007 3,94512332 5 0,04266739 3,98779071 6 0,00985826 3,99764897 7 0,00195235 3,99960132 8 0,00033832 3,99993964 9 5,2112E-05 3,99999175 10 7,2242E-06 3,99999897

Table 5: The sum of  for given that

 Replacing in the form 0 1 1 1 1,60943791 2,60943791 2 1,2951452 3,90458311 3 0,69481859 4,5994017 4 0,27956685 4,87896855 5 0,0899891 4,96895765 6 0,02413864 4,99309629 7 0,00554995 4,99864624 8 0,00111654 4,99976278 9 0,00019967 4,99996244 10 3,2135E-05 4,99999458

Table 6: The sum of  for given that

 Replacing in the form 0 1 1 1 1,79175947 2,79175947 2 1,605201 4,39696047 3 0,95871136 5,35567183 4 0,42944504 5,78511687 5 0,15389244 5,93900931 6 0,04595637 5,98496569 7 0,01176325 5,99672894 8 0,00263461 5,99936356 9 0,00052451 5,99988807 10 9,398E-05 5,99998205

Table 7: The sum of  for given that

Analysing the results of each of the  values we can notice similar results to previous exercises. To make a deeper analysis of data I will plot the three tables on a single graph, showing only the results of the third column. Each of the final results will be represented by colour:  ,

Graph 3: Shows the relation between and , given and

Looking in analysis, we can see that each of the replaced values of increase exponentially to a certain point where they are stabilized. The point in which they stabilize is the replaced value of  and again, the reason for this is that all of the curves are convergent. So to generalize, we can confirm that the value of  tends to the replaced value of  in such way that the value of  tends to .

Conclusion

and  in the sequence, including the result up to 10 and even the infinite, it will always give us the same value that was replaced in the progression, bur this time it will be . Even though, as the limits fluctuate it is said that it is approximately , in other words it is precise but not accurate.

To test the validity of my general statement I will take other values of  and .

I will consider the sequence with the expression of . So the sequence will look like this. If it is correct the result will be 2401

, ,  ,   ,

Hence the table with the summation will look like this:

Table 18: The sum of  for given that  and

Hence the table should look like this:

 24 0.00394168 2401 25 0.00122722 2401 26 0.000367395 2401 27 0.000105914 2401 28 2.94426e-05 2401 29 7.90245e-06 2401

And therefore the graph should correspond to this:

Graph 5: Shows the relation between  vs

The greatest limitation that it has is the fact that you limit them of the sum not always are the same so the result is not always exact.

The general this general statement was obtained by means of the analysis of the tables and the graphic ones that were used throughout the exercise. When replacing by different values for  and , the general proposal was fulfilled so that the terms gave the result of . In the first place for the  values when , and was fulfilled later for the values of  when values of , which demonstrated that the results could not be the same. After I did a deeper analysis of graphs and tables I was able to confirm the general statement by the importance of the relation that there is between the value of  that is replaced and the respective value for .

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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