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Infinite summation SL Portfolio type I. Concerning the portfolio, the evaluation is composed on the sum of the series

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Introduction

Infinite summation – SL Portfolio type I

image00.png

Name: Abdelrahman Lotfy Mosalam

Presented to: Mr. Hamed Mokhtar

Date: 11/10/2011

Class: DP2

The infinite series is almost considered as the main tool in calculus, it has different utilizes. It guesses the behavior of functions, investigates differential equations and also it's used in numerical analysis. Beside these uses in math, the infinite series may be used in physics and economics as well.  

Concerning the portfolio, the evaluation is composed on the sum of the series image01.pngimage01.png below, where:

 t0 = 1, t1 = image75.png, t2 = image80.png, t3 =image85.png, … , tn =image90.png....

 for the value image94.pngimage94.png in the series above, the task gave a notation for the factorial image02.pngimage02.png, described as :

image07.png

image12.pngimage12.png

image24.pngimage24.png (Given)

To find out the general statement that represents the infinite sum of this sequence,  it's always required to determine the sum image03.pngimage03.png of the first image39.pngimage39.png terms of the infinite sequence for image51.pngimage51.png, for where image61.pngimage61.png

image73.pngimage73.png

Using the notations given mentioned and the sequence values, we concluded this table containing the variation of image74.pngimage74.png and it's effect on the image03.pngimage03.png



Note for ALL tables and graphs:

The only controlled variable in the table is the image76.pngimage76.png

All of the answers below will be corrected to 6 decimal places

Domain of n is image51.pngimage51.png

Graphs are all done using Microsoft Excel

Relation between image77.pngimage77.png and image78.pngimage78.png                 (image79.pngimage79.png

image13.png

image81.png

image82.png

image11.png

0

1

2

1.000000

1

1

2

1.693147

2

1

2

1.933374

3

1

2

1.988878

4

1

2

1.998496

5

1

2

1.999829

6

1

2

1.999983

7

1

2

1.999999

8

1

2

2.000000

9

1

2

2.000000

10

1

2

2.000000

image83.png

I can notice right now from this plot, that the image03.pngimage03.png

...read more.

Middle

3.000000

 Again, let’s plot a graph representing the relation between image03.pngimage03.png and image39.pngimage39.png.

image97.png

        This plot insure the proof that the image03.pngimage03.png value is increasing while the  image98.pngimage98.png is also increasing. This time, the graph would not go furtherimage99.pngimage99.png. It's noticeable that the asymptote of this graph is image100.pngimage100.png The limitation that I found in the first trial can be valid, by the statement that asimage22.pngimage22.png, the values of image23.pngimage23.png ,  in this case it is also 3.

at the moment, I will evaluate the sum image03.pngimage03.png, when image04.pngimage04.png, and different values of image05.pngimage05.png. The different values of image05.pngimage05.png will be as following 0.5, 0.1, 25. First two trials with rational numbers then the third with a greater number than all the others trials. Before starting this evaluation, we can agree that the image06.pngimage06.png cannot be a zero or a negative value, , because the domain of  image05.pngimage05.png  is for positive numbers only as (image08.pngimage08.png),also we cannot take a zero beside (ln) as for instance: image09.pngimage09.png simply

Let’s try now the what we planned for, so first try the variable image05.pngimage05.png as image10.pngimage10.png

Relation between image11.pngimage11.png and image13.pngimage13.pngimage14.pngimage14.png

image15.png

image16.png

image17.png

image18.png

0

1

0.5

1.000000

1

1

0.5

0.306853

2

1

0.5

0.547079

3

1

0.5

0.491575

4

1

0.5

0.501193

5

1

0.5

0.499860

6

1

0.5

0.500014

7

1

0.5

0.499999

8

1

0.5

0.500000

9

1

0.5

0.500000

10

1

0.5

0.500000

image19.png

        From this trial, the image20.pngimage20.png isn't behaving exponentially, the relation between image21.pngimage21.png rise and fall up and down, and then it reaches image20.pngimage20.png. However, my notice is that it still stands as when as image22.pngimage22.png, the values of image23.pngimage23.png. In this graph, also I noticed that aimage25.pngimage25.png, the graph swing up and down. The graph may fluctuate up and down and varies below the asymptote, but will never intersect with the asymptote.

Now let's go for a small rational number to see what happens…

Relation between image11.pngimage11.png and image13.pngimage13.pngimage26.pngimage26.png

image15.png

image16.png

image17.png

image18.png

0

1

0.1

1

1

1

0.1

-1.302585093

2

1

0.1

1.348363962

3

1

0.1

-0.68631463

4

1

0.1

0.484940519

5

1

0.1

-0.05444241

6

1

0.1

0.152553438

7

1

0.1

0.084464073

8

1

0.1

0.104061768

9

1

0.1

0.099047839

10

1

0.1

0.100202339

...read more.

Conclusion

image52.pngimage52.png will increase at an exponential rate. So we can now be sure of our statement saying that: image57.pngimage57.png.

To more insure this statement and go for it in more depth, I tried another trial using image58.pngimage58.png, I observed that there is a quite big difference concerning the result, between the algebraically and geographically ones or it appears also on the tables, as by looking at the graph,image59.pngimage59.pngimage60.pngimage60.png , but on the calculator I found 3 at the power of 5 equals 243.000000. This may be considered as a limitation of the investigation. But the situation was treated by the fact that we are only using the 9th term of the sequence, but if we increase the n value the difference will decrease until it reaches the same result, and the n that equals the algebraically result, is the final term in the sequence...

Now finally getting the final general statements discovered in the investigation, by all the trials above analysis we can finally say that:

Firstlyas the image42.pngimage42.png value approaches image62.pngimage62.png, the values of image63.pngimage63.png will approach image64.pngimage64.pngThis relation can be represented as:  

image65.png

The domain of the general statement is:

image67.pngimage66.png

So now we can say that:

image68.png

With respect to domain listed above...

Finally let's test the validity of the general statement by using variables a and x

image69.png

image70.png

image15.png

image17.png

image16.png

image71.png

image72.png

1.000000

0.500000

6.000000

0.895887

2.000000

0.500000

6.000000

0.447946

3.000000

0.500000

6.000000

0.149313

4.000000

0.500000

6.000000

0.037328

5.000000

0.500000

6.000000

0.007466

0.007466

END  OF THE INVESTIGATION...

Abdelrahman Lotfy          Greenland School          IA Maths type I, infinite summation

...read more.

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