1, , , …
=1 and =1
=
This table shows the same result of throughout the increasing value of n which is 1. Therefore, with the table that is consistent, it can be assumed that the line of will be a straight line, with no changing values.
This graph displays the value of =1 and =1 and . Since the value of is 1 there is no change in the graph because the graph starts at 1 on the y-axis and the horizontal asymptote is also 1, therefore the line is straight.
This is another example of the same previous sequence where =1 and =5
=
This table shows the increasing value of with the value of increasing till it reached to 10. approaches 5, but never actually reaches 5, which is discovered that 5 is a horizontal asymptote.
This graph displays the line where =1 and =5 and the value of approaches the horizontal asymptote, which is 5. This creates a longer and steeper line compare to the previous graph since it reaches towards the horizontal asymptote of 5 in the same values of .
Now other values with be investigated to ensure the idea about acting as the horizontal asymptote of the sum of the infinite sequence and the disparity of the graph due to the large value of a, =1 and =50
This table shows the increasing values of 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, =, =, =…
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.
This graph displays the large increasing values of in the beginning that approaches a horizontal asymptote at 32. The value of is and there is exponential growth until the value of n is 7, which is when the rate of increase of begins to slow down and it never actually reaches 32.
Another example where a=2 and x=3, (2, 3)
This table shows the value of being 2 again while the value of is now 3 and the value of still increases until 9. Examining the table, increases quickly until the value of is about 4, where the value of continues to stay in the range of 7 never reaching 8, since the horizontal asymptote is 8.
This graph displays a quick increase of and later on levels out when n is 7, but never exceeds 8 because 8 is the horizontal asymptote. At the later values of (8 and 9) it is visible that the line being shown as a straight line proving that 8 is the horizontal asymptote and will never exceed.
Here is another example of the sequence taking the same concept where ,
This table shows the large changes in the value of throughout,
while quickly reaching the horizontal asymptote of 128, but this horizontal asymptote is not visible in the small values of .
This graph displays = 2 and = 7 and, which creates the horizontal asymptote at 128 as is 128. Unusually, this graph does not display the horizontal asymptote because the end of this line is not yet straight, so it can be assumed that it will straighten more as the value of gets closer to 128. Thus from the three different values of it is visible that changes the horizontal asymptote because it is how is rooted, creating a larger or smaller horizontal asymptote.
Next the same sequence will be used, = 1, =, =, =, (3,) with changing values of to determine the affect has on the sequence with = 3, which is also changed.
This is the first example where = 3 and = 3, (3, 5)
This table shows the fast increasing values of towards 27 then later on skirling around 25 to 26 when n is 6, it can be estimated that it will never exceed 27 because it quickly increases and then the rate of increase slows around 25 with increased values of .
The graph displays the value of = 3 and = 3 and . The line begins to flatten around 27, which is the horizontal asymptote, and it can be stated that is 27. Since the horizontal asymptote is smaller, the exponential growth is shorter and gets straighter at top of the line because there is a smaller rate of increase of as it approaches 27.
Next other values of and will be investigated to guarantee the general statement is valid, and , (4, 2)
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
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.
