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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.

Extracts from this document...

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

that approach to 50, but never actually reaches 50. As the table is shown, it is visible that at each value of  the value of  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.

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

as it approaches 27.

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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