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Infinite Summation Internal Assessment The idea of this internal assessment is to investigate the effect changing the value of x and a have on the graph of the general sequence given.

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Introduction

Shin Park

January 20, 2011

Lancaster 3rd

Infinite Summation Internal Assessment

        The idea of this internal assessment is to investigate the effect changing the value of x and a have on the graph of the general sequence given. In able to observe and display the following graph and tables, Microsoft Excel took action during this investigation. It is also vital to take note of the scatter plot lines with connected dots of the values of image00.pngimage00.png at the different values of image01.pngimage01.png is used for visual assistance to help understand the affect image03.pngimage03.png andimage64.pngimage64.png. The notation of image00.pngimage00.png is the sum of the terms and image01.pngimage01.png is the number of the term in the sequence.

The sequence that will be examine to determine the effect of different values of x and a is  

image06.pngimage06.png=1, image10.pngimage10.png=image11.pngimage11.png, image12.pngimage12.png= image13.pngimage13.png, image14.pngimage14.png= image31.pngimage31.png …, image16.pngimage16.png= image42.pngimage42.png …

The first variation of the sequence that will be observed is, 1,image45.pngimage45.png,image51.pngimage51.png,image54.pngimage54.pngwith x = 1 and a = 2.

n

Sn

0

1.000000

1

1.693147

2

1.933374

3

1.988878

4

1.998496

5

1.999829

6

1.999983

7

1.999999

8

2.000000

9

2.000000

10

2.000000

image00.pngimage00.png=

The table shows the value ofimage58.pngimage58.png, which is sum of the terms in the sequence, which gradually increase to 2, but never passes 2 regardless of the value of n increasing.

image59.png

The graph above displays x=1 a=2 in the sequence and how its sum affects this line. image60.pngimage60.png which is the x-axis, shows that image00.pngimage00.png never passing through 2 because 2 is the horizontal asymptote, thus even if the value of n increases, image00.pngimage00.png would still be smaller than 2.

The next sequence will be a similar concept of the previous one, but the change in values of x=1 and a=3 which is 1, image61.pngimage61.png, image62.pngimage62.png, image63.pngimage63.png

n

Sn

0

1.000000

1

2.098612

2

2.702087

3

2.923082

4

2.983779

5

2.997115

6

2.999557

7

2.999940

8

2.999993

9

2.999999

10

3.000000

image00.pngimage00.png=

...read more.

Middle

image00.png that approach to 50, but never actually reaches 50. As the table is shown, it is visible that at each value of image01.pngimage01.png the value of image00.pngimage00.png increases at larger numbers than the other tables before, it is due to the sum of the sequence is reaching a high value of asymptote in the same amount of time.

image08.png

        This graph demonstrates the values of image03.pngimage03.png=1 and image04.pngimage04.png=50 and image09.pngimage09.png. Also image00.pngimage00.png approaches the horizontal asymptote of 50, which is the value of image04.pngimage04.png. Compare to the graphs before, this graph has a different form of a line due to larger value of a.

From the investigation of the different values of image04.pngimage04.png with the value of image03.pngimage03.png being 1 in this general sequence,

image06.pngimage06.png=1, image10.pngimage10.png=image11.pngimage11.png, image12.pngimage12.png=image13.pngimage13.png, image14.pngimage14.png=image15.pngimage15.png …, image16.pngimage16.png=image17.pngimage17.png.

It can be stated that the value of image03.pngimage03.png is 1 while the horizontal asymptote is determined by the value of a in the sum of this sequence as it approaches infinite. Since image03.pngimage03.png remained as 1, the only changing variable in this whole sequence was image04.pngimage04.png and the value of image04.pngimage04.png was the number where the horizontal asymptote was present on the y-axis.

Now the sum of the infinite sequence image18.pngimage18.pngmust be determined by using a different formula unlike the previous one,

image06.pngimage06.png= 1, image10.pngimage10.png=image19.pngimage19.png, image12.pngimage12.png=image20.pngimage20.png, image14.pngimage14.png=image21.pngimage21.png

a

x

n

t

Sn

2

5

0

1

1

1

3.465736

4.46573590

2

6.005663

10.47139858

3

6.938014

17.40941216

4

6.011331

23.42074285

5

4.166737

27.58747977

6

2.406802

29.99428140

7

1.19162

31.18590123

8

0.51623

31.70213118

9

0.198791

31.90092192

This is an example of image22.pngimage22.png(image23.pngimage23.png, where image24.pngimage24.png(2, 5) which means image04.pngimage04.png= 2 and image03.pngimage03.png= 5

As the table is shown, the value of image04.pngimage04.png is defined as 2 instead of 1 looking back at the section before, and the value of image03.pngimage03.png being 5 with the increasing value of n until 9. The value of image00.pngimage00.png continues to increase, but never exceeds 32.

image25.png

...read more.

Conclusion

image00.pngimage00.png as it approaches 27.

Next other values of image03.pngimage03.png and image04.pngimage04.png will be investigated to guarantee the general statement is valid, image47.pngimage47.png and image48.pngimage48.png, image24.pngimage24.png(4, 2)

a

x

n

t

Sn

4

2

0

1.000000

1.000000

1

2.772589

3.772589

2

3.843624

7.616213

3

3.552263

11.168476

4

2.462241

13.630717

5

1.365356

14.996073

6

0.630929

15.627002

7

0.249901

15.876903

8

0.086609

15.963512

9

0.026681

15.990193

The table above shows a fast increasing value of the sum of the sequence until n=6 as it approaches 16 at a slow rate, which 16 is the horizontal asymptote since image49.pngimage49.png= 16.

image50.png

This graph displays the value of image47.pngimage47.png and image48.pngimage48.png and image26.pngimage26.png. There is exponential growth in the sum of the sequence until the value of image01.pngimage01.png is about 3, where the rate of line of increase begins to slow since it is approaching the horizontal asymptote of 16.

Another example is with image52.pngimage52.png and image48.pngimage48.png, image24.pngimage24.png(5, 2) and using the same sequence as before

a

x

n

t

Sn

5

2

0

1.000000

1.000000

1

3.218876

4.218876

2

5.180581

9.399457

3

5.558549

14.958005

4

4.473070

19.431075

5

2.879651

22.310726

6

1.544873

23.855599

7

0.710394

24.565993

8

0.285834

24.851826

9

0.102229

24.954056

This table shows the increasing value of the sum of the sequence approaching 25 as the value of image01.pngimage01.png increases, but looks as it will never exceed 25, due to the fact that 25 is the horizontal asymptote.

image53.png

This graph displays the value of the sum of this sequence when image52.pngimage52.png and image48.pngimage48.png and image55.pngimage55.png Since in this sequence image56.pngimage56.png is image57.pngimage57.png= 25, the horizontal asymptote is then 25, which is why the line has exponential growth until it approaches 25 creating an almost straight line.

After the investigation of the infinite sequence,

image35.pngimage35.png= 1, image36.pngimage36.png=image37.pngimage37.png, image38.pngimage38.png=image39.pngimage39.png, image40.png=image41.pngimage41.png

Where image22.pngimage22.png(image23.pngimage23.png, with the changing values of image03.pngimage03.png and image04.pngimage04.png, it is clear that image56.pngimage56.png is the horizontal asymptote of the sum of the terms in this sequence as image01.pngimage01.png approaches infinite. The limitations of this general statement could vary since image01.pngimage01.png was only investigated until certain numbers not infinity, therefore it was examination and judgment that created the general statements about the horizontal asymptotes that affect the graph of the sum of the infinite sequence.  

...read more.

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