- Level: International Baccalaureate
- Subject: Maths
- Word count: 1457
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: ____________________________
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
- 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
4,878967
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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