- Level: International Baccalaureate
- Subject: Maths
- Word count: 2332
SL Type I Mathematics Portfolio
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Introduction
Infinite Summation
SL Type I Mathematics Portfolio
Infinite series are among the most powerful and useful tools that you will encounter in calculus. They are among the major tools used in analyzing differential equations, developing methods of numerical analysis, defining new functions, and estimating behaviour of functions, and much more. The use of infinite series can be found in a variety of fields, such as electronics engineering, micro-economics, mathematics, and physics.
In this Mathematical portfolio, the investigation of the sum of infinite sequence, , where: t0 = 1, t1 = , t2 = , t3 =, … , tn =.
For a positive integer, factorial , written as , is the product of all of the positive integers less than or equal to .
.
Furthermore, (by definition)
This can be proofed with the equation
The above sequence is called an infinite sequence, because the three dots at the end of the sequence indicate that the sequence continues indefinitely. However, the following sequence is called a finite sequence because it has a finite number of terms, . After determine this sequence as an infinite sequence, we should determine whether or not this sequence is convergent or divergent. This is important because if this sequence diverges, the general statement would be .
Middle
1
15
1.000000
1
1
15
3.708050
2
1
15
7.374818
3
1
15
10.684749
4
1
15
12.925613
5
1
15
14.139288
6
1
15
14.687070
7
1
15
14.898987
8
1
15
14.970723
9
1
15
14.992307
10
1
15
14.998153
From this set of data, I can see that the value is approaching 15, as the value approaches 10. The asymptote in this plot is . The line cannot cross or intersect with .
Relation between and
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 |
From this set of data, I can see that instead of increasing exponential until , the relation between Relations between fluctuates up and down, and then it reaches . However, my observation still stands as when as , the values of . In this graph, I noticed something that has not happened with a, the graph fluctuates up and down. I observe that when , the graph dips below the asymptote at . This is due to the fact that , the answer is negative. When the is an even number, the negative values cancel each other out, thus producing a relatively larger number. However, when the is an odd number, the would be relatively low, and often a negative number. Thus, the graph would fall below the asymptote when . The graph will fluctuates above and below the asymptote, but never intersecting the asymptote.
Relation between and
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 |
This set of data sees a lot more fluctuation than the last set of data when This is also the first time that is less than 0, when . This is because that the produce a number that is too small. The graph would continue to fluctuate until . Even though this graph has a big fluctuation, it can still take the observation that as , the values of
After investigating the cases above, it is certain that when , the general statement that represents the infinite sum of the general sequence is as , the values of . This can be written as the following:
Right now, we will expand this investigation to determine the sum of the infinite sequence
We can define that as the sum of the first n terms, for various values of .
Let a =2. Calculate for various positive values of .
Using technology, plot the relation between and . Describe the relationship.
Relation between and
0.01 | 1.006956 | 1.006956 |
0.1 | 1.071773 | 1.071773 |
0.25 | 1.189207 | 1.189207 |
0.5 | 1.414214 | 1.414214 |
1 | 2.000000 | 2.000000 |
1.5 | 2.828427 | 2.828427 |
2 | 3.999992 | 4.000000 |
2.5 | 5.656775 | 5.656854 |
3 | 7.999488 | 8.000000 |
4 | 15.990193 | 16.000000 |
Conclusion
This general statement is an example of the Maclaurin series. It is a series that expresses a function in terms of an infinite power series whose nth coefficient is the nth derivative of f(x), evaluated at . The Maclaurin series The Maclaurin Series can be written as this :
There are also some common functions of the Maclaurin series. The expansion of by the Maclaurin series is written as followed:
This validates that my general statement - is true.
Here are more values of to test the validity of the general statement.
0.000000 | 0.500000 | -2.000000 | 1.000000 | |
1.000000 | 0.500000 | -2.000000 | 2.386294 | |
2.000000 | 0.500000 | -2.000000 | 3.347200 | |
3.000000 | 0.500000 | -2.000000 | 3.791233 | |
4.000000 | 0.500000 | -2.000000 | 3.945123 | |
5.000000 | 0.500000 | -2.000000 | 3.987791 | |
6.000000 | 0.500000 | -2.000000 | 3.997649 | |
7.000000 | 0.500000 | -2.000000 | 3.999601 | |
8.000000 | 0.500000 | -2.000000 | 3.999940 | |
9.000000 | 0.500000 | -2.000000 | 3.999992 | |
10.000000 | 0.500000 | -2.000000 | 3.999999 | |
11.000000 | 0.500000 | -2.000000 | 4.000000 | |
12.000000 | 0.500000 | -2.000000 | 4.000000 | 4.000000 |
0.000000 | 2.000000 | -0.500000 | 1.000000 | |
1.000000 | 2.000000 | -0.500000 | 0.653426 | |
2.000000 | 2.000000 | -0.500000 | 0.713483 | |
3.000000 | 2.000000 | -0.500000 | 0.706545 | |
4.000000 | 2.000000 | -0.500000 | 0.707146 | |
5.000000 | 2.000000 | -0.500000 | 0.707104 | |
6.000000 | 2.000000 | -0.500000 | 0.707107 | |
7.000000 | 2.000000 | -0.500000 | 0.707107 | |
8.000000 | 2.000000 | -0.500000 | 0.707107 | |
9.000000 | 2.000000 | -0.500000 | 0.707107 | 0.707107 |
From the last two tables, we witness that even though the x value is negative, it still does not change the general statement.
0.000000 | 0.100000 | 2.000000 | 1.000000 | |
1.000000 | 0.100000 | 2.000000 | -3.605170 | |
2.000000 | 0.100000 | 2.000000 | 6.998626 | |
3.000000 | 0.100000 | 2.000000 | -9.278803 | |
4.000000 | 0.100000 | 2.000000 | 9.461280 | |
5.000000 | 0.100000 | 2.000000 | -7.798974 | |
6.000000 | 0.100000 | 2.000000 | 5.448760 | |
7.000000 | 0.100000 | 2.000000 | -3.266678 | |
8.000000 | 0.100000 | 2.000000 | 1.750331 | |
9.000000 | 0.100000 | 2.000000 | -0.816800 | |
10.000000 | 0.100000 | 2.000000 | 0.365408 | |
11.000000 | 0.100000 | 2.000000 | -0.129526 | |
12.000000 | 0.100000 | 2.000000 | 0.060412 | |
13.000000 | 0.100000 | 2.000000 | -0.006872 | |
14.000000 | 0.100000 | 2.000000 | 0.015260 | |
15.000000 | 0.100000 | 2.000000 | 0.008465 | |
16.000000 | 0.100000 | 2.000000 | 0.010421 | |
17.000000 | 0.100000 | 2.000000 | 0.009891 | |
18.000000 | 0.100000 | 2.000000 | 0.010027 | |
19.000000 | 0.100000 | 2.000000 | 0.009994 | |
20.000000 | 0.100000 | 2.000000 | 0.010001 | |
21.000000 | 0.100000 | 2.000000 | 0.010000 | |
22.000000 | 0.100000 | 2.000000 | 0.010000 | 0.010000 |
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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