Infinite summation portfolio. A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation
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Introduction
International Baccalaureate
Math Standard Level Internal Assessment
Portfolio Type: I
Portfolio Title: Infinite Summation
Due date: 9th of December, 2011
Teacher: Mr. Peter Vassilev
School: The American College of Sofia
Candidate Name: Rami Meziad
Candidate number: 002368-008
Examination Session: May 2012
Introduction:
A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation [3] is a series which continues indefinitely. The Taylor's theorem [1] and the Euler-Maclaurin's formula [2] will help us solve our given infinite summation, which is: 
,
,
,
,
And by adding different values for x and a, we will be able to find a general pattern in which the sequences tends to move with. And this is mainly what this portfolio will ask us to do.
Method:
For our sequence, which is:
, we have to substitute in the case where x = 1 and a = 2. After that, we have to calculate the first n terms which happen to be eleven to fulfill the given condition
.
So after substitution we get 
Now let's calculate for n, when 
:
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In fact, t9and t10 are not equal to 0, but since we have to take our answers correct to six decimal places, we can't see the real values. However, the numbers become so small, that they become insignificant, or in other words they are equal to 0.
Now, we need to find the sum of Sn :
Now, using Excel 2010, let's plot the relation between Sn
Middle
We see that 
and the asymptote (which is also approximately the value of S10) of Sn are the same (in this case the asymptote is 7). It seems that 
, however we will try with our last value, and see if this is true:
For 
:
Now we find S10 for
Now, we draw for the last time the relation between Sn and n for (with Excel 2010 again):
Sn
We note that as 
, 
As we were observing all of those different values for a, we’ve noticed that as 
, 
, and also 
, and if we make an approximate estimation of the Sn, we could easily say that 
. And this is the general statement that applies in our case. However, theoretically, this only holds true when 
.
We will examine for different values of x, and then tell what is the true general statement for any values for x and a. For now, it is 
.
Now, in order to get a general statement for 
we will have to change the values for a and x, and then find a pattern.
We will start with 
, and for x we will try various positive values.
To compare right the results we’ll look correct to 6 decimal places again:
:
T8
It’s obvious that the graph keeps the same way as the one described in all previous graphs. We can see that at the 5th
Conclusion
a = π.
...read more.
For 
T8
For , the statement is still the same. The last check we can do is with a negative value of x, as for a we cannot do that because of the limitation if the ln function.
For 
T8
Clearly it is the first graph that is distinguishable and doesn’t follow the common behavior as well as these are the first values that don’t follow the statement Sn = an. The values as well as the graph seem to oscillate under and above the 0 with increasing amplitude of oscillation as much as n approaches infinity.
As a conclusion we can say that the general statement Sn = an fulfils quite well the presented infinite summation, including the first case when we stated that Sn = a, which is a particular case when x = 1, so it is according to a1 = a.
The statement should consider the following limitations:
a ≠ 0, x ≠ 0 – as this will give us a single value “0” and the graph will be a straight line on the x axis.
a > 0, x > 0 – as a > 0 is the natural limitation of the ln, and x > 0 shows an oscillating graph.
Works Cited:
[1] "Taylor's Theorem." Wikipedia, the Free Encyclopedia. Web. 09 Dec. 2011. <http://en.wikipedia.org/wiki/Taylor's_theorem>.
[2]"Euler–Maclaurin formula." Wikipedia, the Free Encyclopedia. Web. 09 Dec. 2011.<http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula>
[3] "Series (mathematics)." Wikipedia, the Free Encyclopedia. Web. 09 Dec. 2011. <http://en.wikipedia.org/wiki/Series_(mathematics)>.
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