Next, we can consider the same sequence with , but changing the variable to . The result of the sequence is as following:
, , , , …
Now that we know the sequence we can now get the sum of the first terms of the above sequence for . Like the sequence before, I will use Microsoft Excel in order to get the table and therefore get the sum of all terms.
The table is as following:
Table 3: Shows the sum of the first 10 terms for
After analysing the previous results the sum of the first terms of the above sequence for is 2,99999992.
Likewise, we can represent the relationship between and in a graph by replacing the values of in the y-axis and the values of in the x-axis as seen in the previous sequence.
The table should look like this:
Table 4: Shows the values of the graph of against of
After using the computer programme Graphmatica, the result of the graph is this:
Graph 2: Shows the relation between and in
Same as the previous sequence, we can observe that the values increase exponentially but they never reach 3. At this point they stabilize. Hence, we can conclude that this sequence is also convergent. This happens because the values are getting smaller rather than being bigger due to the factorial notation in the denominator. As the denominator gets bigger we obtain results with smaller numbers. From this information we can say that the values tend to move to 3 if the variable is , likewise the way approaches to .
Next, we will consider the same general sequence where in such way that:
, , , , …
In this sequence we will replace different values for variable . In my case I will use 3 different values that can replace : 4, 5 and 6. For each of them I will calculate the sum of the first terms of the above sequence for . To do this I will use again the help of Microsoft Excel as done previously in other sequences.
Table 5: The sum of for given that
Table 6: The sum of for given that
Table 7: The sum of for given that
Analysing the results of each of the values we can notice similar results to previous exercises. To make a deeper analysis of data I will plot the three tables on a single graph, showing only the results of the third column. Each of the final results will be represented by colour: ,
Graph 3: Shows the relation between and , given and
Looking in analysis, we can see that each of the replaced values of increase exponentially to a certain point where they are stabilized. The point in which they stabilize is the replaced value of and again, the reason for this is that all of the curves are convergent. So to generalize, we can confirm that the value of tends to the replaced value of in such way that the value of tends to .
After looking previous investigations we are able to obtain a general statement that represents the infinite sum of the general sequence. The general statement is:
This statement comes from all the results obtained in previous investigations due to the fact that every time that we replace the value of in the sequence, including the result up to 10 and even the infinite, it will always give us the same value that was replaced in the progression. Even though, as the limits fluctuate it is said that it is approximately , in other words it is precise but not accurate.
In order to proof the general statement I will do the same process as before but with a bigger number in such way that . So the first term of the second column of the table should look like this:
Completing the table with the help of the spread sheet in Excel it should look like this:
Table 8: The sum of for given that
As we can see form the general statement, the value that replaces in the function will be result of the sum from 0 to . Also it is important to notice that the sequence reaches to number 20 meaning that from summation 18 to , the result will be 20.
Subsequent to expanding this investigation we will consider again our original sequence:
, , , , …
In this part of the exercise we will consider the expression , which represents the sum of the first terms, for various values of and .
In first we will have the progression in which the variable , hence . To find out the sum of the first terms we are going to calculate different values of and calculate their results with Excel. The 4 different values of are 0.01, 0.6, 3, and 5. The results of their summation are in the following tables:
Table 9: Table 10:
Table 11: Table 12:
Based on the results from the tables and the sum until the term 9 we can now plot a graph that represents the relation with variable . The graph will contain the following data
Table 13: Data of graph vs
Graph 3: Shows the relation between vs
Analysing the graph we can see that points increase exponentially as the curve fit line can show us. We can see that the values of have increased by the expression from the original value of . For example if , then the value of , which is equal to 4.
The next exercise is that we will have the same progression, but this time we will change variable to , hence . To find out the sum of the first terms we are going to calculate different values of like we did before in Excel. The 4 different values of are 1, 2, 4, and 8. The results of their summation are in the following tables:
Table 14: Table 15:
Table 15: Table 16:
Based on the results from the tables and the sum until the term 9 we can now plot a graph that represents the relation with variable like on the previous exercise. The graph will contain the following data:
Table 17: Data of graph vs
Graph 4: Shows the relation between vs
Results on this graph are not surprising and actually they show the same values but with the variable . We can see that the values of have increased by the expression from the original value of . For example if , then the value of , which is equal to 5.
Based on the previous exercises we are now able to establish the general statement for as approaches . The general statement is as following
This statement comes from all the results obtained in previous investigations due to the fact that every time that we replace the value of and in the sequence, including the result up to 10 and even the infinite, it will always give us the same value that was replaced in the progression, bur this time it will be . Even though, as the limits fluctuate it is said that it is approximately , in other words it is precise but not accurate.
To test the validity of my general statement I will take other values of and .
I will consider the sequence with the expression of . So the sequence will look like this. If it is correct the result will be 2401
, , , , …
Hence the table with the summation will look like this:
Table 18: The sum of for given that and
Hence the table should look like this:
And therefore the graph should correspond to this:
Graph 5: Shows the relation between vs
The greatest limitation that it has is the fact that you limit them of the sum not always are the same so the result is not always exact.
The general this general statement was obtained by means of the analysis of the tables and the graphic ones that were used throughout the exercise. When replacing by different values for and , the general proposal was fulfilled so that the terms gave the result of . In the first place for the values when , and was fulfilled later for the values of when values of , which demonstrated that the results could not be the same. After I did a deeper analysis of graphs and tables I was able to confirm the general statement by the importance of the relation that there is between the value of that is replaced and the respective value for .
