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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.

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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:                ____________________________ 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 of the first 10 terms of the above sequence for           After I calculated the sums for I am going to represent the relation between on a graph:

 n 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 , 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 , as n approaches

.

Let

therefore the sequence of 1.  will look in the following way:           After I calculated the sums for I am going to represent the relation between on a graph:

 n 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 , 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 , as n approaches

.

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 , 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 , as n approaches

.

Let

therefore the sum of will look the following way:           After I calculated the sums for I am going to represent the relation between on a graph:

 n 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 , 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 , 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 ( ). In order to continue with finding the general formula, first I have to simplify the original sum of infinite sequences: . In order to do this I will replace (xlna) with m, therefore this sequence will look in the following: Hence the sum of these infinite terms will be: However on power series expansion’s side my general statement will have the following form: From here follows that the general statement for the infinite sequences is  I concluded that this is the general statement, on the basis of the following facts: as 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 , where In order to find the general statement of these infinite sequences I will define as the sum of the first n terms for various values of a and x.

First I will find , 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 2x and x I need to plot on a graph:

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

 X (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

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 increases also, which support my general statement and proves the statement that as the value of , 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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