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SL Type I Mathematics Portfolio

Extracts from this document...

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, image00.pngimage00.png,  where:       t0 = 1, t1 = image41.png, t2 = image85.png, t3 =image111.png, … , tn =image116.png.

        For a positive integerimage118.pngimage118.png, factorial image11.pngimage11.png, written as image01.pngimage01.png, is the product of all of the positive integers less than or equal to image11.pngimage11.png.

image21.png
.        
image23.pngimage23.png

        Furthermore, image29.pngimage29.png (by definition)

This can be proofed with the equation image37.pngimage37.png

image46.png

image50.png

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 image54.pngimage54.png  is called a finite sequence because it has a finite number of terms, image11.pngimage11.png. 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  image69.pngimage69.png.

...read more.

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

image20.png

        From this set of data, I can see that the image03.pngimage03.png value is approaching 15, as the image11.pngimage11.png value approaches 10. The asymptote in this plot is image22.pngimage22.png. The line cannot cross or intersect with image22.pngimage22.png.

Relation between image03.pngimage03.png and image11.pngimage11.pngimage24.png

image16.png

image17.png

image18.png

image19.png

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

image25.png

        From this set of data, I can see that instead of increasing exponential until image26.pngimage26.png, the relation between Relations betweenimage27.pngimage27.png fluctuates up and down, and then it reaches image26.pngimage26.png. However, my observation still stands as when as image07.pngimage07.png, the values of image08.pngimage08.png. In this graph, I noticed something that has not happened with aimage28.pngimage28.png, the graph fluctuates up and down. I observe that when image30.pngimage30.png, the graph dips below the asymptote at image31.pngimage31.png. This is due to the fact that image32.pngimage32.png, the answer is negative. When the image33.pngimage33.pngis an even number, the negative values cancel each other out, thus producing a relatively larger number. However, when the image04.pngimage04.png is an odd number, the image34.pngimage34.png would be relatively low, and often a negative number. Thus, the graph would fall below the asymptote when image35.pngimage35.png. The graph will fluctuates above and below the asymptote, but never intersecting the asymptote.

Relation between image03.pngimage03.png and image11.pngimage11.pngimage36.pngimage36.png

image16.png

image17.png

image18.png

image19.png

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

image38.png

        This set of data sees a lot more fluctuation than the last set of data when image39.pngimage39.png This is also the first time that image40.pngimage40.pngis less than 0, when image42.pngimage42.png. This is because that theimage43.pngimage43.png produce a number that is too small. The graph would continue to fluctuate until image44.pngimage44.png. Even though this graph has a big fluctuation, it can still take the observation that as image07.png, the values of image45.pngimage45.png

        After investigating the cases above, it is certain that when image09.pngimage09.png, the general statement that represents the infinite sum of the general sequence is as image07.pngimage07.png, the values of image08.pngimage08.png. This can be written as the following:

image47.png

Right now, we will expand this investigation to determine the sum of the infinite sequence

image48.png

We can define that image49.pngimage49.pngas the sum of the first n terms, for various values of image51.pngimage51.png.

Let a =2. Calculate image52.pngimage52.png for various positive values of image53.pngimage53.png.

Using technology, plot the relation between image52.pngimage52.png and image53.pngimage53.png. Describe the relationship.

Relation between image03.pngimage03.png and image11.pngimage11.pngimage55.pngimage55.png

image17.png

image56.png

image57.png

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

...read more.

Conclusion

image93.pngimage93.png, there are other times when the image03.pngimage03.png value can be off. As noticed in the example, when image76.pngimage76.png which is not accurate with the function image75.pngimage75.png. This is because I am only taking the 9th term of this sequence. If I take image94.pngimage94.png, the answer would be closer to image95.pngimage95.png This pattern can be noted that as the image96.pngimage96.pngthe number of terms that required for image97.pngimage97.png would also increase. This can be further mentioned that as the image98.pngimage98.png, the number of terms that required for image97.pngimage97.png will increase as well.

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 image100.pngimage100.png. The Maclaurin series The Maclaurin Series can be written as this :

image101.png

There are also some common functions of the Maclaurin series. The expansion of image93.pngimage93.pngby the Maclaurin series is written as followed:

image102.png

This validates that my general statement - image103.pngimage103.png is true.

Here are more values of image51.pngimage51.png to test the validity of the general statement.

image16.png

image18.png

image17.png

image104.png

image105.png

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

image16.png

image18.png

image17.png

image104.png

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.

image16.png

image18.png

image17.png

image104.png

image107.png

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

...read more.

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