- Level: International Baccalaureate
- Subject: Maths
- Word count: 1662
Infinite Summation - IB Math SL Mathematics Portfolio 2011
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Introduction
Mathematics SL Portfolio Type II | 2012 | 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: ____________________________
Class: ____________________________
Index Number: ____________________________
Subject Teacher: ____________________________
Table of Contents
Table of Contents
Introduction
Part I
Irrational number: When x=1 and a=2
Negative number: When x=1 and a=-3
0 and 1:
Non-integer: When x=1 and a=0.87
Proving the general statement
Part II
T9 (2,x)
T9 (3,x)
Validating the General Statement
a= 6
When a=2/3
Scope
Limitations
For when a is a negative number:
For when a is equal to 0:
For when a is equal to 1:
Conclusion
Introduction
A summation is the operation of adding a sequence of numbers, with the result being their sum total^{[1]}. An infinite series is essentially a series with an infinite number of terms. Due to the indeterminable nature of all the terms, mathematical analysis is required to fully understand the series and its properties. However, that is not to say that infinite series are not used often. In fact, they are regularly used in the fields of physics and computer science. ^{[2]}In this portfolio, we will be investigating the sum of infinite sequences tnwhere
t0=1, t1=(xlna)1, t2=(xlna)22!, t3=(xlna)33!,…,tn=(xlna)nn!
By observing various trends within the series, I will come up with some general statements and will test the validity of said statements with the aid of some examples. Then, I will try to link the patterns discovered to some pre-established mathematical concepts, such as the Maclaurin series.
Middle
1
#NUM!
#NUM!
2
#NUM!
#NUM!
n | tn | sn |
0 | #NUM! | #NUM! |
1 | 0.000000 | #NUM! |
2 | 0.000000 | #NUM! |
As we can see, the above tables prove that the sequence is only applicable to numbers for which the value of [xln(1)]n is defined.
Non-integer: When x=1 and a=0.87
n | tn | sn |
0 | 1.000000 | 1.000000 |
1 | -0.139262 | 0.860738 |
2 | 0.009697 | 0.870435 |
3 | -0.000450 | 0.869985 |
4 | 0.000016 | 0.870000 |
5 | 0.000000 | 0.870000 |
6 | 0.000000 | 0.870000 |
7 | 0.000000 | 0.870000 |
8 | 0.000000 | 0.870000 |
9 | 0.000000 | 0.870000 |
10 | 0.000000 | 0.870000 |
Once again, we see that even for non-integers, while the value of Sn does fluctuate a little, it does eventually reach 0.87 as n tends towards ∞.
From the above examples, we can conclude that a general statement can be made regarding the series, i.e. when x=1, Sn →a as n→∞.
Proving the general statement
Now let us try to prove the general statement for the series for when x=1, using something more concrete than mere observation. To do this, let us make use of the Maclaurin series^{[3]}, explained in detail in the scope section of the portfolio. We will be using the exponential form of the series:
ey=1+y1+y22!+y33!+…+y22!+…
Substituting the values with those of the sequence above:
exlna=1+xlna1+(xlna)22!+(xlna)33!+…+(xlna)nn!+…
When x=1:
elna=1+lna1+(lna)22!+(lna)33!+…+(lna)nn!+…
Let elna=p
ln both sides:
lnalne=lnp
∴lna=lnp
∴p=elna
=a
Part II
To arrive at the above general statement, we varied the value of a. Now, let us try varying the value of x to see if we can arrive at a similar general statement. Let us take Tn (a,x) as the sun of the first n terms for different values of a and x. Now, let us consider the case where a=2.
T9 (2,x)
x | T9(2,x) |
1 | 2 |
2 | 3.99994 |
3 | 7.997485 |
4 | 15.96351 |
5 | 31.70213 |
6 | 62.30535 |
7 | 120.4657 |
8 | 227.9637 |
9 | 420.6994 |
10 | 755.6926 |
From the above values of the above table, we can see for each value of x, the value of T9 (2,x) approaches the value of ax. For example, when x=5, T9 (2,x) tends towards 32, that is 25. However, starting from when x=6, the value of T9 (2,x) begins to deviate significantly from the value of ax.
T9 (3,x)
Now let us look at T9 (3,x)
T9 (3,x)
x | T9(3,x) |
1 | 2.999993 |
2 | 8.995812 |
3 | 26.81482 |
4 | 78.11916 |
5 | 217.4715 |
6 | 569.0338 |
7 | 1390.257 |
8 | 3174.043 |
9 | 6802.981 |
10 | 13771.39 |
Conclusion
For when a is a negative number:
tn=(xlna)nn!
t0=(xln(-1))00!
Let x=1
xln-1=ln-11
=0
∴t0=000!
The above value is undefined, thus the series does not work for negative numbers.
For when a is equal to 0:
tn=(xlna)nn!
t0=(xln0)00!
Again,
t0=000!
Which is undefined, thus the series does not work for when a is equal to 0 either.
For when a is equal to 1:
tn=(xlna)nn!
t0=(xln1)00!
Again,
t0=000!
Thus, we can prove that the series expansion does not work for when a=0, a=1 or
a<0.
Conclusion
The Mathematics SL Type 1 Portfolio has helped me in many ways. As someone who is not all that great at mathematics, it has helped me tremendously to learn how much the use of technology can aid in the study of mathematics. I also learnt a great deal about infinite summation, the Taylor and Maclaurin series as well as about how they help us even in or daily life. Also, the portfolio has helped me analyze and rethink many of my approaches to mathematics thus far. Thus, from this portfolio I have not only learnt about infinite summation, but a great deal about myself and the influence mathematics has on the world around me as well.
Page
[1]http://en.wikipedia.org/wiki/Summation
[2]http://en.wikipedia.org/wiki/Series_(mathematics)
[3]http://www.intmath.com/series-expansion/2-maclaurin-series.php
[4]http://en.wikipedia.org/wiki/Taylor_series
[5]http://en.wikipedia.org/wiki/Taylor_series
[6]http://mathworld.wolfram.com/MaclaurinSeries.html
[7]http://www.intmath.com/series-expansion/2-maclaurin-series.php
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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