- Level: International Baccalaureate
- Subject: Maths
- Word count: 1688
Infinite summation SL Portfolio type I. Concerning the portfolio, the evaluation is composed on the sum of the series
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Introduction
Infinite summation – SL Portfolio type I
Name: Abdelrahman Lotfy Mosalam
Presented to: Mr. Hamed Mokhtar
Date: 11/10/2011
Class: DP2
The infinite series is almost considered as the main tool in calculus, it has different utilizes. It guesses the behavior of functions, investigates differential equations and also it's used in numerical analysis. Beside these uses in math, the infinite series may be used in physics and economics as well.
Concerning the portfolio, the evaluation is composed on the sum of the series below, where:
t0 = 1, t1 = , t2 = , t3 =, … , tn =....
for the value in the series above, the task gave a notation for the factorial , described as :
(Given)
To find out the general statement that represents the infinite sum of this sequence, it's always required to determine the sum of the first terms of the infinite sequence for , for where
Using the notations given mentioned and the sequence values, we concluded this table containing the variation of and it's effect on the
Note for ALL tables and graphs:
The only controlled variable in the table is the
All of the answers below will be corrected to 6 decimal places
Domain of n is
Graphs are all done using Microsoft Excel
Relation between and (
0 | 1 | 2 | 1.000000 |
1 | 1 | 2 | 1.693147 |
2 | 1 | 2 | 1.933374 |
3 | 1 | 2 | 1.988878 |
4 | 1 | 2 | 1.998496 |
5 | 1 | 2 | 1.999829 |
6 | 1 | 2 | 1.999983 |
7 | 1 | 2 | 1.999999 |
8 | 1 | 2 | 2.000000 |
9 | 1 | 2 | 2.000000 |
10 | 1 | 2 | 2.000000 |
I can notice right now from this plot, that the
Middle
3.000000
Again, let’s plot a graph representing the relation between and .
This plot insure the proof that the value is increasing while the is also increasing. This time, the graph would not go further. It's noticeable that the asymptote of this graph is The limitation that I found in the first trial can be valid, by the statement that as, the values of , in this case it is also 3.
at the moment, I will evaluate the sum , when , and different values of . The different values of will be as following 0.5, 0.1, 25. First two trials with rational numbers then the third with a greater number than all the others trials. Before starting this evaluation, we can agree that the cannot be a zero or a negative value, , because the domain of is for positive numbers only as (),also we cannot take a zero beside (ln) as for instance: simply
Let’s try now the what we planned for, so first try the variable as
Relation between and
0 | 1 | 0.5 | 1.000000 |
1 | 1 | 0.5 | 0.306853 |
2 | 1 | 0.5 | 0.547079 |
3 | 1 | 0.5 | 0.491575 |
4 | 1 | 0.5 | 0.501193 |
5 | 1 | 0.5 | 0.499860 |
6 | 1 | 0.5 | 0.500014 |
7 | 1 | 0.5 | 0.499999 |
8 | 1 | 0.5 | 0.500000 |
9 | 1 | 0.5 | 0.500000 |
10 | 1 | 0.5 | 0.500000 |
From this trial, the isn't behaving exponentially, the relation between rise and fall up and down, and then it reaches . However, my notice is that it still stands as when as , the values of . In this graph, also I noticed that a, the graph swing up and down. The graph may fluctuate up and down and varies below the asymptote, but will never intersect with the asymptote.
Now let's go for a small rational number to see what happens…
Relation between and
0 | 1 | 0.1 | 1 |
1 | 1 | 0.1 | -1.302585093 |
2 | 1 | 0.1 | 1.348363962 |
3 | 1 | 0.1 | -0.68631463 |
4 | 1 | 0.1 | 0.484940519 |
5 | 1 | 0.1 | -0.05444241 |
6 | 1 | 0.1 | 0.152553438 |
7 | 1 | 0.1 | 0.084464073 |
8 | 1 | 0.1 | 0.104061768 |
9 | 1 | 0.1 | 0.099047839 |
10 | 1 | 0.1 | 0.100202339 |
Conclusion
To more insure this statement and go for it in more depth, I tried another trial using , I observed that there is a quite big difference concerning the result, between the algebraically and geographically ones or it appears also on the tables, as by looking at the graph, , but on the calculator I found 3 at the power of 5 equals 243.000000. This may be considered as a limitation of the investigation. But the situation was treated by the fact that we are only using the 9th term of the sequence, but if we increase the n value the difference will decrease until it reaches the same result, and the n that equals the algebraically result, is the final term in the sequence...
Now finally getting the final general statements discovered in the investigation, by all the trials above analysis we can finally say that:
Firstlyas the value approaches , the values of will approach This relation can be represented as:
The domain of the general statement is:
So now we can say that:
With respect to domain listed above...
Finally let's test the validity of the general statement by using variables a and x…
1.000000 | 0.500000 | 6.000000 | 0.895887 | ||
2.000000 | 0.500000 | 6.000000 | 0.447946 | ||
3.000000 | 0.500000 | 6.000000 | 0.149313 | ||
4.000000 | 0.500000 | 6.000000 | 0.037328 | ||
5.000000 | 0.500000 | 6.000000 | 0.007466 | 0.007466 |
END OF THE INVESTIGATION...
Abdelrahman Lotfy Greenland School IA Maths type I, infinite summation |
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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