- Level: International Baccalaureate
- Subject: Maths
- Word count: 2076
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 at the different values of is used for visual assistance to help understand the affect and. The notation of is the sum of the terms and 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
=1, =, = , = …, = …
The first variation of the sequence that will be observed is, 1,,,with 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 |
=
The table shows the value of, 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.
The graph above displays x=1 a=2 in the sequence and how its sum affects this line. which is the x-axis, shows that never passing through 2 because 2 is the horizontal asymptote, thus even if the value of n increases, 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, , , …
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 |
=
Middle
This graph demonstrates the values of =1 and =50 and . Also approaches the horizontal asymptote of 50, which is the value of . 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 with the value of being 1 in this general sequence,
=1, =, =, = …, =.
It can be stated that the value of is 1 while the horizontal asymptote is determined by the value of a in the sum of this sequence as it approaches infinite. Since remained as 1, the only changing variable in this whole sequence was and the value of was the number where the horizontal asymptote was present on the y-axis.
Now the sum of the infinite sequence must be determined by using a different formula unlike the previous one,
= 1, =, =, =…
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 (, where (2, 5) which means = 2 and = 5
As the table is shown, the value of is defined as 2 instead of 1 looking back at the section before, and the value of being 5 with the increasing value of n until 9. The value of continues to increase, but never exceeds 32.
Conclusion
Next other values of and will be investigated to guarantee the general statement is valid, and , (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 = 16.
This graph displays the value of and and . There is exponential growth in the sum of the sequence until the value of 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 and , (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 increases, but looks as it will never exceed 25, due to the fact that 25 is the horizontal asymptote.
This graph displays the value of the sum of this sequence when and and Since in this sequence is = 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,
= 1, =, =, =…
Where (, with the changing values of and , it is clear that is the horizontal asymptote of the sum of the terms in this sequence as approaches infinite. The limitations of this general statement could vary since 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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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