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In this portfolio, I will determine the general sequence tn with different values of variables to find the formula to count the sum of the infinite sequence.

Extracts from this document...

Introduction

002329/

Anglo-Chinese School (Independent)

INFINITE SUMMATION

“I undersigned, hereby declare that the following course work is all my own work and that I worked independently on it”

__________________

Name:                                Pranav Sharma _______________        

Class:                                5.16 Ephesians_______________        

Index Number:                ____________________________

Subject Teacher:                ____________________________

image12.jpg

Introduction

  • Technology used:
  • Microsoft word 2010
  • Microsoft Excel 2010

In this portfolio, I will determine the general sequence tn with different values of variables to find the formula to count the sum of the infinite sequence.


Let

, therefore I will calculate the image00.png

 of the first 10 terms of the above sequence for

image01.png

image02.png

image03.png

image03.png

image03.png

image03.png

image03.png

image03.png

image03.png

image03.png

image03.png

After I calculated the sums for image04.png

 I am going to represent the relation between image00.png

 on a graph:

n

image00.png

1

1,693147

2

1,933373

3

1,988877

4

1,998495

5

1,999828

6

1,999982

7

1,999997

8

1,999998

9

1,999998

10

1,999998

From this graph we can see that as image00.png

, the value of n increases as well, but it does not exceed 2, thus the greatest value of this relationship will be 2 and therefore the domain of this relationship is image00.png

, as n approaches

.

Let

therefore the sequence of image04.png

  1.  will look in the following way:

image01.png

image02.png

image05.png

image05.png

image05.png

image05.png

image05.png

image05.png

image05.png

image05.png

image05.png

After I calculated the sums for image04.png

 I am going to represent the relation between image00.png

 on a graph:

n

image00.png

1

2,098612

2

2,702086

3

2,923080

4

2,983776

5

2,997112

6

2,999553

7

2,999936

8

2,999988

9

2,999994

10

2,999994

From this graph we can see that as image00.png

, the value of n increases as well, but it does not exceed 3, thus the greatest value of this relationship will be 3 and therefore the domain of this relationship is image00.png

, as n approaches

.

...read more.

Middle

4

4,878967

5

4,968956

6

4,993095

7

4,998645

8

4,999761

9

4,999961

10

4,999993

From this graph we can see that as image00.png

, the value of n increases as well, but it does not exceed 5, thus the greatest value of this relationship will be 5 and therefore the domain of this relationship is image00.png

, as n approaches

.

Let

 therefore the sum of image04.png

will look the following way:

image01.png

image02.png

image05.png

image05.png

image05.png

image05.png

image06.png

image05.png

image05.png

image05.png

image05.png

After I calculated the sums for image04.png

 I am going to represent the relation between image00.png

 on a graph:

n

image00.png

1

2,945910

2

4,839193

3

6,067246

4

6,664666

5

6,897171

6

6,972577

7

6,993539

8

6,998637

9

6,999740

10

6,999954

From this graph we can see that as image00.png

, the value of n increases as well, but it does not exceed 7, thus the greatest value of this relationship will be 7 and therefore the domain of this relationship is image00.png

, as n approaches

.

From all of the calculations which I performed now I can find a general statement for finding the sequence of this sum (image00.png

). In order to continue with finding the general formula, first I have to simplify the original sum of infinite sequences: image07.png

. In order to do this I will replace (xlna) with m, therefore this sequence will look in the following:

image08.png

Hence the sum of these infinite terms will be:

image09.png

However on power series expansion’s side my general statement will have the following form:

image10.png

From here follows that the general statement for the infinite sequences is

image00.png

image00.png

I concluded that this is the general statement, on the basis of the following facts: as image00.png

 increases the value of n will increase also

 Now I want to expand my investigation in order to find a general statement which represents the infinite sum of the infinite sequence image00.png

, where

image07.png

In order to find the general statement of these infinite sequences I will define image00.png

 as the sum of the first n terms for various values of a and x.

First I will find image00.png

, where I have a=2 and I calculate x for 1,2,3,4,5,6,7,8,9,10

Therefore I will calculate the sum for the first nine terms:

In order to comment on the relation between image00.png

2x and x I need to plot on a graph:

x

image00.png

(2,x)

1

1,999999

2

3,999939

3

7,997485

4

15,963511

5

31,702131

6

62,305349

7

120,465724

8

227,963677

9

502,560081

10

755,692613

Looking at the graph I can conclude that as x increases, the value of image00.png

 increases as well.

Now I am going to perform the same calculations as above, but with values of a=3 and x=1,2,3,4,5,6,7,8,9,10

Therefore I will calculate the sum for the first nine terms:

In order to comment on the relation between image00.png

(3,x) and x I need to plot on a graph:

X

image00.png

(2,x)

1

2,999992

2

6,798587

3

26,814822

4

78,119155

5

217,471546

6

569,033836

7

1969,842574

8

3060,893218

9

6802,981103

10

13771,38794

...read more.

Conclusion

x=2

x=3

x=4

x=8

x=6

a=2

4

8

16

64

256

a=3

9

27

81

729

6561

a=4

16

64

256

4096

65536

a=5

25

125

625

15625

390625

a=6

36

216

1296

46656

1679616

From the table which I made I can clearly see that as value of a and x increases the value of image00.png

 increases also, which support my general statement and proves the statement that as the value ofimage00.png

, the values of a and x increases too. Therefore my general statement can fit any values of a and x.

  • We know the range of values Tn (a,x).
  • We know how Tn (a,x) changes when x changes.
  • We know the domain for a and x: positive numbers.
  • It’s easy to find the infinite sum, just by setting values for a and x.
  • We know this is power series expansion.

What I did to find out this statement is just calculating, and while doing this, I have been realising, step by step, some signs that suggest about the Tn (a,x), like range, sign and simplest formula.

After doing this portfolio, I learned several things, such as

  • using mathematical technology on computer, which I did not know before;
  • constructing the parts of the work so that it looks logically;
  • using appropriate language when doing mathematical big work;
  • realising subtleties from graph/plot rather than from statistics as before I was used to;
  • how to find the sum of infinite general sequence.

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...read more.

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