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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

Extracts from this document...

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: image00.png,image01.png,image106.png,image203.png,image292.pngimage303.png

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:

image313.png, 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 conditionimage322.png.

So after substitution we get image332.png

Now let's calculate for n, when image281.png:

image02.png

image13.png

image24.png

image34.png

image45.png

image55.png

image65.png

image75.png

image86.png

image96.png

image107.png

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 :

image46.png

image126.png

image136.png

image146.png

image155.png

image165.png

image174.png

image184.pngimage193.pngimage204.pngimage213.png

Now, using Excel 2010, let's plot the relation between Sn

...read more.

Middle

We see that image30.png 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 image31.png, however we will try with our last value, and see if this is true:

For image32.png :

image33.png

image35.png

image36.png

image37.png

image38.png

image39.png

image40.png

image41.png

image42.png

image43.png

image44.png

Now we find S10 for image32.png

image46.png

image47.png

image48.png

image49.png

image50.png

image51.png

image52.png

image53.pngimage54.pngimage56.pngimage57.png

Now, we draw for the last time the relation between Sn and n for image32.png (with Excel 2010 again):

Sn

image58.png

We note that as image59.png, image60.png

As we were observing all of those different values for a, we’ve noticed that as image61.png, image62.png, and also image63.png, and if we make an approximate estimation of the Sn, we could easily say that image64.png. And this is the general statement that applies in our case. However, theoretically, this only holds true when image66.png.

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 image67.png.

Now, in order to get a general statement for image68.png we will have to change the values for a and x, and then find a pattern.

We will start with image69.png, and for x we will try various positive values.

To compare right the results we’ll look correct to 6 decimal places again:

image70.png:

image33.png

image71.png

image72.png

image73.png

image74.png

image76.png

image77.png

image78.png

image79.png

image80.png

image81.png

image82.png

image83.png

image84.png

image85.png

image87.png

image88.pngimage89.png

T8

image90.png

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

...read more.

Conclusion

a = π.

For image229.pngimage250.png

image33.png

image251.png

image253.png

image254.png

image255.png

image256.png

image257.png

image258.png

image259.png

image80.png

image260.png

image261.png

image263.png

image264.png

image265.png

image266.pngimage267.pngimage268.png

T8

image269.png

For image229.pngimage250.png, 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 image270.pngimage272.png

image33.png

image273.png

image274.png

image275.png

image276.png

image277.png

image278.png

image279.png

image280.png

image80.png

image282.png

image283.png

image284.png

image285.png

image286.png

image287.pngimage288.pngimage289.png

T8

image290.png

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)>.

...read more.

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